flexible codebook design for limited feedback systems via sequential smooth optimization on the grassmannian manifold grassmannian quantization codebooks play a central role in a number of limited feedback schemes for single and multi user multiple input multiple output mimo communication systems in practice it is often desirable that these codebooks possess additional properties that facilitate their implementation beyond the provision of good quantization performance although some good codebooks exist their design tends to be a rather intricate task the goal of this paper is to suggest a flexible approach to the design of grassmannian codebooks based on sequential smooth optimization on the grassmannian manifold and the use of smooth penalty functions to obtain additional desirable properties as one example the proposed approach is used to design rank 2 codebooks that have a nested structure and elements from a phase shift keying psk alphabet in some numerical comparisons codebooks designed using the proposed methods have larger minimum distances than some existing codebooks and provide tangible performance gains when applied to a simple mimo downlink scenario with zero forcing beamforming per user unitary beamforming and rate control pu 2rc and block diagonalization signalling furthermore the proposed approach yields codebooks that attain desirable additional properties without incurring a substantial degradation in performance

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

the even orlicz minkowski problem the classical minkowski problem leads to the lp minkowski problem and now to the orlicz minkowski problem existence is demonstrated for the even orlicz minkowski problem a byproduct is a new approach to the solution of the classical minkowski problem

an island strategy for memetic discrete tomography reconstruction in this paper we present a parallel island model memetic algorithm for binary discrete tomography reconstruction that uses only four projections without any further a priori information the underlying combination strategy consists in separated populations of agents that evolve by means of different processes agents progress towards a possible solution by using genetic operators switch and a particular compactness operator a guided migration scheme is applied to select suitable migrants by considering both their own and their sub population fitness that is from time to time we allow some individuals to transfer to different subpopulations the benefits of this paradigm were tested in terms of correctness robustness and time of the reconstruction by considering publicly available datasets of images to tackle the so called stability problem we considered the case of noisy projections along four directions to simulate an instrumental error results show that the proposed method decreases the reconstruction error for all classes of images with respect to a serial implementation recently proposed by the authors and that such reconstruction error is almost invariant with respect to the number of demes moreover the computation time of the proposed parallel memetic algorithm scales in a quasi linear manner with respect to the demes number and is invariant with respect to the used number of migrations

reconstruction of tomographic images using analog projections and the digital radon transform abstract the digital radon transform drt can be adapted to reconstruct images from analog projection data this new technique is a variation of the conventional back projection method it requires no pre filtering of the projection data straightforward 1d linear interpolation and some simple sorting of projection samples the drt enables the use of a form of block data copy for the reconstruction which is fast in comparison to the usual methods of back projection to obtain reconstructed images of high quality further intrinsic interpolation is required the reconstructed image size has to be several times larger than the number of projection samples we describe an algorithm to convert analog projection data into a form suitable to apply the drt we compare the performance of the standard drt and a hybrid version of the drt to some conventional reconstruction algorithms

sections of convex bodies with symmetries abstract in this paper we study how certain symmetries of convex bodies affect their geometric properties in particular we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations generalizing complex and quaternionic convex bodies we conduct a systematic study of sections of bodies with symmetries of this type with the emphasis on problems of the busemann petty type and hyperplane inequalities the main role belongs to the class of intersection bodies with symmetries

single valued representation of set valued mappings ii application to differential inclusions we consider a set valued mapping q t x the first argument ranging through a measurable space and the second through a space with a different structure topological metric differentiable etc we are interested in representing q by a single valued mapping with properties compatible with the structures i e in the existence of a topological space z and a mapping f t x z measurable in t continuous in x z and depending on x as dictated by the structure of the space of x s say continuous lipschitz or differentiable in x etc and such that f t x z q t x for all t x approximate representations i e those for which the latter equality holds approximately of convex valued mappings are also considered and certain applications mainly to differential inclusions are discussed

limited feedback unitary precoding for spatial multiplexing systems multiple input multiple output mimo wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity spatial multiplexing is a common space time modulation technique for mimo communication systems where independent information streams are sent over different transmit antennas unfortunately spatial multiplexing is sensitive to ill conditioning of the channel matrix precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter which is often not realistic this correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter the index of the optimal precoder is conveyed from the receiver to the transmitter over a low delay feedback link criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a rayleigh fading matrix channel the design criteria are shown to be equivalent to packing subspaces in the grassmann manifold using the projection two norm and fubini study distances simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback

intersection bodies and generalized cosine transforms the paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis a new concept of intersection body that unifies some known classes of geometric objects is introduced a parallel between trace theorems in function theory restriction onto lower dimensional subspaces of the spherical radon transforms and the generalized cosine transforms and sections of intersection bodies is established new integral formulas for different classes of cosine transforms are obtained and examples of intersection bodies are given we also revisit some known facts in this area and give them new simple proofs

the busemann petty problem via spherical harmonics abstract the busemann petty problem asks whether symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume the solution has recently been completed and the answer is affirmative if n 4 and negative if n 5 in this article we present a short proof of the affirmative result and its generalization using the funk hecke formula for spherical harmonics

if you can hide behind it can you hide inside it let k and l be compact convex sets in rn suppose that for a given dimension 1 d n 1 every d dimensional orthogonal projection of l contains a translate of the corresponding projection of k does it follow that the original set l contains a translate of k in other words if k can be translated to hide behind l from any perspective does it follow that k can hide inside l a compact convex set l is defined to be d decomposable if l is a direct minkowski sum affine cartesian product of two or more convex bodies each of dimension at most d a compact convex set l is called d reliable if whenever each d dimensional orthogonal projection of l contains a translate of the corresponding d dimensional projection of k it must follow that l contains a translate of k it is shown that for 1 d n 1 1 d decomposability implies d reliability 2 a compact convex set l in rn is d reliable if and only if for all m d 2 no m unit normals to regular boundary points of l form the outer unit normals of an m 1 dimensional simplex 3 smooth convex bodies are not d reliable 4 a compact convex set l in rn is 1 reliable if and only if l is 1 decomposable i e a parallelotope 5 a centrally symmetric compact convex set l in rn is 2 reliable if and only if l is 2 decomposable however there are non centered 2 reliable convex bodies that are not 2 decomposable as a result of 5 above the only reliable centrally symmetric covers in r3 from the perspective of 2 dimensional shadows are the affine convex cylinders prisms however in dimensions greater than 3 it is shown that 3decomposability is only sufficient and not necessary for l to cover reliably with respect to 3 shadows even when l is assumed to be centrally symmetric consider two compact convex subsetsk and l of n dimensional euclidean space suppose that for a given dimension 1 d 0 is sufficiently small but no longer fits inside d the same construction works for any set k inscribed in d and having strictly less than unit diameter another received by the editors june 4 2009 2000 mathematics subject classification primary 52a20 c 2011 daniel a klain

high snr analysis for mimo broadcast channels dirty paper coding versus linear precoding in this correspondence we compare the achievable throughput for the optimal strategy of dirty paper coding dpc to that achieved with suboptimal and lower complexity linear precoding techniques zero forcing and block diagonalization both strategies utilize all available spatial dimensions and therefore have the same multiplexing gain but an absolute difference in terms of throughput does exist the sum rate difference between the two strategies is analytically computed at asymptotically high snr furthermore the difference is not affected by asymmetric channel behavior when each user has a different average snr weighted sum rate maximization is also considered in the process it is shown that allocating user powers in direct proportion to user weights asymptotically maximizes weighted sum rate

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

quantization bounds on grassmann manifolds and applications to mimo communications the grassmann manifold gn p l is the set of all p dimensional planes through the origin in the n dimensional euclidean space ln where l is either r or c this paper considers the quantization problem in which a source in gn p l is quantized through a code in gn q l with p and q not necessarily the same the analysis is based on the volume of a metric ball in gn p l with center in gn q l and our chief result is a closed form expression for the volume of a metric ball of radius at most one this volume formula holds for arbitrary n p q and l while previous results pertained only to some special cases based on this volume formula several bounds are derived for the rate distortion tradeoff assuming that the quantization rate is sufficiently high the lower and upper bounds on the distortion rate function are asymptotically identical and therefore precisely quantify the asymptotic rate distortion tradeoff we also show that random codes are asymptotically optimal in the sense that they achieve the minimum possible distortion in probability as n and the code rate approach infinity linearly finally as an application of the derived results to communication theory we quantify the effect of beamforming matrix selection in multiple antenna communication systems with finite rate channel state feedback

image reconstruction from linograms implementation and evaluation the notion of a linogram corresponds to the notion of a sinogram in the conventional representation of projection data for image reconstruction in the sinogram points which correspond to rays through a fixed point in the cross section to be reconstructed all fall on a sinusoidal curve in the linogram however these points fall on a straight line the implementation of a novel image reconstruction method using this property is discussed the implementation is of order n sup 2 log n where n is proportional to the number of pixels on a side of the reconstruction region it is demonstrated that the performance of the algorithm is superior to that of the filter backprojection method in computational speed on realistic size problems and is equivalent to filtered backprojection in accuracy of reconstruction

qam and psk codebooks for limited feedback mimo beamforming this paper considers the problem of beamforming in multiple input multiple output mimo wireless systems assuming perfect channel state information at the receiver the choice of the beamforming vector is made possible through a noiseless limited rate feedback of one or more bits per coefficient to the transmitter this paper proposes the use of beamforming codebooks based on quadrature amplitude modulation qam and phase shift keying psk constellations which essentially eliminates the need for storage of the codebook we show that such codebooks perform arbitrarily close to the perfect feedback case as the constellation size increases and that full diversity order is achieved we demonstrate an equivalence between the beamforming codebook search problem with that of noncoherent sequence detection based on this we propose fast beamforming vector search algorithms monte carlo simulations are presented to show that the performance is comparable to the best known codebooks and that the search complexity can be reduced by several orders of magnitude

the isoperimetrix in the dual brunn minkowski theory abstract we introduce the dual isoperimetrix which solves the isoperimetric problem in the dual brunn minkowski theory we then show how the dual isoperimetrix is related to the isoperimetrix from the brunn minkowski theory

systematic design of unitary space time constellations we propose a systematic method for creating constellations of unitary space time signals for multiple antenna communication links unitary space time signals which are orthonormal in time across the antennas have been shown to be well tailored to a rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients the signals can achieve low probability of error by exploiting multiple antenna diversity because the fading coefficients are not known the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximum euclidean distance norm our construction begins with the first signal in the constellation an oblong complex valued matrix whose columns are orthonormal and systematically produces the remaining signals by successively rotating this signal in a high dimensional complex space this construction easily produces large constellations of high dimensional signals we demonstrate its efficacy through examples involving one two and three transmitter antennas

dual orlicz brunn minkowski theory abstract in this paper a dual orlicz brunn minkowski theory is presented an orlicz radial sum and dual orlicz mixed volumes are introduced the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality are established the variational formula for the volume with respect to the orlicz radial sum is proved the equivalence between the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality is demonstrated orlicz intersection bodies are defined and the orlicz busemann petty problem is posed

kerdock codes for limited feedback precoded mimo systems codebook based limited feedback is a practical way to obtain partial channel state information at the transmitter in a precoded multiple input multiple output wireless system in this correspondence we propose a systematic codebook design the proposed kerdock codebook consists of multiple mutually unbiased unitary basis matrices with quaternary alphabet and the identity matrix we propose to derive the beamforming and precoding codebooks from this base codebook eliminating the requirement to store multiple codebooks furthermore we show that quaternary alphabet facilitates efficient codebook storage and codeword search we derive the chordal distance for two and four antenna codebooks showing that the proposed codebooks compare favorably with prior designs monte carlo simulations compare achievable rates and error rates for different codebooks sizes

an explicit expression for the newton direction on the complex grassmann manifold several important design problems in signal processing for communications can be cast as optimization problems in which the objective is a function of the subspaces spanned by tall complex matrix variables with orthonormal columns such problems can be viewed as optimization problems on the complex grassmann manifold and an effective means for performing this optimization is to use a grassmannian version of newton s method to facilitate the implementation of that method we provide an explicit expression for the grassmannian newton direction for an arbitrary twice differentiable function we also use an example in which the pairwise chordal frobenius norm between subspaces is to be optimized to outline a systematic procedure for obtaining the hessian matrix

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

optimization algorithms exploiting unitary constraints this paper presents novel algorithms that iteratively converge to a local minimum of a real valued function f x subject to the constraint that the columns of the complex valued matrix x are mutually orthogonal and have unit norm the algorithms are derived by reformulating the constrained optimization problem as an unconstrained one on a suitable manifold this significantly reduces the dimensionality of the optimization problem pertinent features of the proposed framework are illustrated by using the framework to derive an algorithm for computing the eigenvector associated with either the largest or the smallest eigenvalue of a hermitian matrix

young s inequality for convolution and its applications in convex and set valued analysis abstract a version of young s inequality for convolution is introduced and employed to some topics in convex and set valued analysis the following problems are considered uniform equivalence of metrics for convex subsets of the euclidean space the regularity of set valued mappings and the continuity of the funk radon transform also an isoperimetric inequality based on young s inequality is introduced

image representation via a finite radon transform a model of finite radon transforms composed of radon projections is presented the model generalizes to finite group projections in the classical radon transform theory the radon projector averages a function on a group over cosets of a subgroup reconstruction formulae that were formally similar to the convolved backprojection ones are derived and an iterative reconstruction technique is found to converge after a finite number of steps applying these results to the group z sub 2 sup p new computationally favorable image representations have been obtained a numerical study of the transform coding aspects is attached

orlicz centroid bodies the sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body from below by the volume of the star body itself is the busemann petty centroid inequality a decade ago the lp analogue of the classical busemannpetty centroid inequality was proved here the definition of the centroid body is extended to an orlicz centroid body of a star body and the corresponding analogue of the busemann petty centroid inequality is established for convex bodies the centroids of the intersections of an origin symmetric body with half spaces form the surface of a convex body this centroid body is a concept that dates back at least to dupin the classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by blaschke and established in a landmark work by petty 53 petty s inequality became known as the busemann petty centroid inequality because in establishing his inequality petty not only made critical use of busemann s random simplex inequality but as petty stated he reinterpreted it see e g the books by gardner 12 leichtweiss 25 schneider 55 and thompson 58 for reference the concept of a centroid body had a natural extension in what became known as the lp brunn minkowski theory and its dual this theory had its origins in the early 1960s when firey see e g schneider 55 introduced his concept of lp compositions of convex bodies three decades later in 34 and 35 these firey minkowski lp combinations were shown to lead to an embryonic lp brunn minkowski theory this theory and its dual has witnessed a rapid growth see e g 1 9 17 23 26 32 34 44 46 48 54 56 57 59 62 the lp analogues of centroid bodies became a central focus within the lp brunn minkowski theory and its dual and establishing the lpanalogue of the busemann petty centroid inequality became a major goal this was accomplished by the authors of the present paper in 37 with an independent approach presented by campi and gronchi 3 the lp centroid bodies quickly became objects of interest in asymptotic mathematics subject classification mr 52a40 research supported in part by nsf grant dms 0706859 proof copy 1 not for distribution 2 erwin lutwak deane yang and gaoyong zhang geometric analysis see e g 10 11 24 49 50 51 52 and even the theory of stable distributions see e g 48 using concepts introduced by ludwig 29 haberl and schuster 21 were led to establish asymmetric versions of the lp busemann petty centroid inequality that for bodies that are not origin symmetric are stronger than the lp busemann petty centroid inequality presented in 37 and 3 the asymmetric inequalities obtained by haberl and schuster are ideally suited for non symmetric bodies this can be seen by looking at the haberl schuster version of the lp analogue of the classical blaschke santalo inequality that was presented in 46 while for origin symmetric bodies the lp extension of 46 does recover the original blaschke santalo inequality as p for arbitrary bodies only the haberl schuster version does so the works of haberl and schuster 21 see also 22 ludwig and reitzner 32 and ludwig 31 have demonstrated the clear need to move beyond the lp brunn minkowski theory to what we are calling an orlicz brunn minkowski theory this need is not only motivated by compelling geometric considerations such as those presented in ludwig and reitzner 32 but also by the desire to obtain sobolev bounds see 22 of a far more general nature this paper is the second in a series intended to develop a few of the elements of an orlicz brunn minkowski theory and its dual here we define the orlicz centroid body establish some of its basic properties and most importantly establish what we call the orlicz busemannpetty centroid inequality for orlicz centroid bodies we consider convex r 0 such that 0 0 this means that must be decreasing on 0 and increasing on 0 we require that one of these is happening strictly so i e is either strictly decreasing on 0 or strictly increasing on 0 the class of such will be denoted by c if k is a star body see section 1 for precise definitions with respect to the origin in rn with volume k and c then we define the orlicz centroid body k of k as the convex body whose support function at x rn is given by h k x inf 0 1 k k x y dy 1 where x y denotes the standard inner product of x and y in rn and the integration is with respect to lebesgue measure in rn when p t t p with p 1 then

a memetic approach to discrete tomography from noisy projections discrete tomography deals with the reconstruction of images from very few projections which is in the general case an np hard problem this paper describes a new memetic reconstruction algorithm it generates a set of initial images by network flows related to two of the input projections and lets them evolve towards a possible solution by using crossover and mutation switch and compactness operators improve the quality of the reconstructed images during each generation while the selection of the best images addresses the evolution to an optimal result one of the most important issues in discrete tomography is known as the stability problem and it is tackled here in the case of noisy projections along four directions extensive experiments have been carried out to evaluate the robustness of the new methodology a comparison with the output of two other evolutionary algorithms and a generalized version of a deterministic method shows the effectiveness of our new algorithm

detecting symmetry in star bodies abstract we use fourier transform techniques to prove a result on detecting symmetry in convex and star bodies with the help of conical sections our methods also allow us to give a new proof of the well known theorem of makai martini and odor about maximal hyperplane sections passing through the same point

the orlicz brunn minkowski inequality abstract the orlicz brunn minkowski theory originated with the work of lutwak yang and zhang in 2010 in this paper we first introduce the orlicz addition of convex bodies containing the origin in their interiors and then extend the l p brunn minkowski inequality to the orlicz brunn minkowski inequality furthermore we extend the l p minkowski mixed volume inequality to the orlicz mixed volume inequality by using the orlicz brunn minkowski inequality

dual mixed volumes and the slicing problem we develop a technique using dual mixed volumes to study the isotropic constants of some classes of spaces in particular we recover strengthen and generalize results of ball and junge concerning the isotropic constants of subspaces and quotients of lp and related spaces an extension of these results to negative values of p is also obtained using generalized intersection bodies in particular we show that the isotropic constant of a convex body which is contained in an intersection body is bounded up to a constant by the ratio between the latter s mean radius and the former s volume radius we also show how type or cotype 2 may be used to easily prove inequalities on any isotropic measure

isotropic fading vector broadcast channels the scalar upper bound and loss in degrees of freedom we propose a scalar upper bound on the capacity region of the isotropic fading vector broadcast channel in terms of the capacity region of a scalar fading broadcast channel the scalar upper bound is applicable to the broad class of isotropic fading broadcast channels regardless of the distribution of the users channel magnitudes the distribution of the additive noise experienced by each user or the amount of channel knowledge available at the receiver using this upper bound we prove the optimality of the alamouti scheme in a broadcast setting extend the recent results on the capacity of nondegraded fading scalar broadcast channels to nondegraded fading vector broadcast channels and determine the capacity region of a fading vector gaussian broadcast channel with channel magnitude feedback we also provide an example of a rayleigh fading broadcast channel with no channel state information available to the receiver csir where the bound on the capacity region obtained by a naive application of the scalar upper bound is provably loose because it fails to account for the additional loss in degrees of freedom due to lack of channel knowledge at the receiver a tighter upper bound is obtained by separately accounting for the loss in degrees of freedom due to lack of csir before applying the scalar upper bound

affine images of isotropic measures necessary and sufficient conditions are given in order for a borel measure on the euclidean sphere to have an affine image that is isotropic a sharp reverse affine isoperimetric inequality for borel measures on the sphere is presented this leads to sharp reverse affine isoperimetric inequalities for convex bodies 2015 international press of boston inc all rights reserved

zero forcing methods for downlink spatial multiplexing in multiuser mimo channels the use of space division multiple access sdma in the downlink of a multiuser multiple input multiple output mimo wireless communications network can provide a substantial gain in system throughput the challenge in such multiuser systems is designing transmit vectors while considering the co channel interference of other users typical optimization problems of interest include the capacity problem maximizing the sum information rate subject to a power constraint or the power control problem minimizing transmitted power such that a certain quality of service metric for each user is met neither of these problems possess closed form solutions for the general multiuser mimo channel but the imposition of certain constraints can lead to closed form solutions this paper presents two such constrained solutions the first referred to as block diagonalization is a generalization of channel inversion when there are multiple antennas at each receiver it is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high snr the second known as successive optimization is an alternative method for solving the power minimization problem one user at a time and it yields superior results in some e g low snr situations both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined in order to accommodate more general scenarios we also propose a framework for coordinated transmitter receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas while the proposed algorithms are suboptimal they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

the log brunn minkowski inequality abstract for origin symmetric convex bodies i e the unit balls of finite dimensional banach spaces it is conjectured that there exist a family of inequalities each of which is stronger than the classical brunn minkowski inequality and a family of inequalities each of which is stronger than the classical minkowski mixed volume inequality it is shown that these two families of inequalities are equivalent in that once either of these inequalities is established the other must follow as a consequence all of the conjectured inequalities are established for plane convex bodies

incremental grassmannian feedback schemes for multi user mimo systems the communication of side information forms a key component of several effective strategies for transmitter adaptation to slowly fading channels when the relevant side information is a subspace the feedback scheme can be viewed as a lossy source compression scheme on the grassmannian manifold memoryless vector quantization on each fading block is a viable compression scheme but it neglects any temporal correlation between the blocks in this paper we propose an incremental approach to grassmannian quantization that takes advantage of temporal correlation the approach leverages existing codebooks for memoryless quantization schemes and employs a quantized form of geodesic interpolation two schemes that implement the principles of the proposed approach are presented in the first scheme the choice of the step size in the incremental update is adapted to a first order gaussmarkov model for the channel which enables the use of higher resolution codebooks in the second scheme a single bit is allocated to the step size which enables adaptation of the step size to the channel realization rather than the channel statistics this provides substantial robustness against mismatches in the model for the temporal correlation a distinguishing feature of the proposed approach is that the direction of the geodesic interpolation is specified implicitly using a point in a conventional codebook as a result the approach has an inherent ability to recover autonomously from errors in the feedback path simulation results demonstrate that these features result in improved performance over some existing schemes in a variety of channel environments

coherence optimization and best complex antipodal spherical codes vector sets with optimal coherence according to the welch bound cannot exist for all pairs of dimension and cardinality if such an optimal vector set exists it is an equiangular tight frame and represents the solution to a grassmannian line packing problem best complex antipodal spherical codes bcascs are the best vector sets with respect to the coherence by extending methods used to find best spherical codes in the real valued euclidean space the proposed approach aims to find bcascs and thereby a complex valued vector set with minimal coherence there are many applications demanding vector sets with low coherence examples are not limited to several techniques in wireless communication or to the field of compressed sensing within this contribution existing analytical and numerical approaches for coherence optimization of complex valued vector spaces are summarized and compared to the proposed approach the numerically obtained coherence values improve previously reported results the drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases

design and analysis of transmit beamforming based on limited rate feedback this paper deals with design and performance analysis of transmit beamformers for multiple input multiple output mimo systems based on bandwidth limited information that is fed back from the receiver to the transmitter by casting the design of transmit beamforming based on limited rate feedback as an equivalent sphere vector quantization svq problem multiantenna beamformed transmissions through independent and identically distributed i i d rayleigh fading channels are first considered the rate distortion function of the vector source is upper bounded and the operational rate distortion performance achieved by the generalized lloyd s algorithm is lower bounded although different in nature the two bounds yield asymptotically equivalent performance analysis results the average signal to noise ratio snr performance is also quantified finally beamformer codebook designs are studied for correlated rayleigh fading channels and a low complexity codebook design that achieves near optimal performance is derived

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

convexity of lp intersection bodies we extend the classical brunn theorem to symmetric moments of convex bodies and use it to prove the convexity of the lp intersection bodies of centered convex bodies

projection functions area measures and the alesker fourier transform abstract dual to koldobsky s notion of j intersection bodies the class of j projection bodies is introduced generalizing minkowski s classical notion of projection bodies of convex bodies a fourier analytic characterization of j projection bodies in terms of their area measures of order j is obtained in turn this yields an equivalent characterization of j projection bodies involving alesker s fourier type transform on translation invariant smooth spherical valuations as applications of these results several basic properties of j projection bodies are established and new non trivial examples are constructed

bounds on packings of spheres in the grassmann manifold we derive the gilbert varshamov and hamming bounds for packings of spheres codes in the grassmann manifolds over r and c asymptotic expressions are obtained for the geodesic metric and projection frobenius chordal metric on the manifold

the lp busemann petty centroid inequality abstract the ratio between the volume of the p centroid body of a convex body k in r n and the volume of k attains its minimum value if and only if k is an origin symmetric ellipsoid this result the lp busemann petty centroid inequality was recently proved by lutwak yang and zhang in this paper we show that all the intrinsic volumes of the p centroid body of k are convex functions of a time like parameter when k is moved by shifting all the chords parallel to a fixed direction the lp busemann petty centroid inequality is a consequence of this general fact

affine inequalities and radial mean bodies we introduce for p 1 the radial pth mean body rpk of a convex body k in en the distance from the origin to the boundary of rpk in a given direction is the pth mean of the distances from points inside k to the boundary of k in the same direction the bodies rpk form a spectrum linking the difference body of k and the polar projection body of k which correspond to p oo and p 1 respectively we prove that rpk is convex when p 0 we also establish a strong and sharp affine inequality relating the volume of rpk to that of rqk when 1 1 and q n it becomes a reverse form of the petty projection inequality proved previously by the second author 0 introduction two important objects in convex geometry are the differ ence body and the polar projection body of a convex body the difference body was studied by minkowski and is ubiquitous in geometry and elsewhere as the vector sum of a set and its reflection in the origin the operation that forms the difference body is essentially that known as central or radial symmetrization and as such finds many applications in mathematical physics and partial differential equations see for example the books of bandle ba and kawohl k projection bodies also originated in the work of minkowski and have found application in the theory of vector valued measures liapounov s theorem the local theory of banach spaces stochastic geometry random determinants hubert s fourth prob lem mathematical economics and other areas see the articles of bourgain and lindenstrauss bl goodey and weil gw and lutwak li the polar pro jection body the polar body of the projection body appears explicitly but very frequently in the more recent literature its behavior under linear transformations often renders it more natural than the projection body itself both the difference body and the polar projection body appear in known affine inequalities the first is an ingredient in the famous rogers shephard inequality corollary 5 8 below an affine inequality relating the volume of a convex body to that of its difference body the second appears in another affine inequality corollary 5 9 first proved by the second author relating the volume of a convex body to that of its polar projection body the latter is a sharp reverse form of the well known petty projection inequality see for example g theorem 9 2 9

covering shadows with a smaller volume for n 2 a construction is given for convex bodies k and l in rn such that the orthogonal projection lu onto the subspace u contains a translate of ku for every direction u while the volumes of k and l satisfy vn k vn l a more general construction is then given for n dimensional convex bodies k and l such that each orthogonal projection l onto a k dimensional subspace contains a translate of k while the mth intrinsic volumes of k and l satisfy vm k vm l for all m k for each k 1 n we then define the collection cn k to be the closure under the hausdorff topology of all blaschke combinations of suitably defined cylinder sets prisms it is subsequently shown that if l cn k and if the orthogonal projection l contains a translate of k for every k dimensional subspace of rn then vn k vn l the families cn k called k cylinder bodies of rn form a strictly increasing chain cn 1 cn 2 cn n 1 cn n where cn 1 is precisely the collection of centrally symmetric compact convex sets in rn while cn n is the collection of all compact convex sets in rn members of each family cn k are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes and members of cn k are shown to satisfy certain geometric inequalities related open questions are also posed

limited feedback unitary precoding for orthogonal space time block codes orthogonal space time block codes ostbcs are a class of easily decoded space time codes that achieve full diversity order in rayleigh fading channels ostbcs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit channel information when channel state information is available at the transmitter though precoding the space time codeword can be used to support different numbers of transmit antennas and to improve array gain unfortunately transmitters in many wireless systems have no knowledge about current channel conditions this motivates limited feedback precoding methods such as channel quantization or antenna subset selection this paper investigates a limited feedback approach that uses a codebook of precoding matrices known a priori to both the transmitter and receiver the receiver chooses a matrix from the codebook based on current channel conditions and conveys the optimal codebook matrix to the transmitter over an error free zero delay feedback channel a criterion for choosing the optimal precoding matrix in the codebook is proposed that relates directly to minimizing the probability of symbol error of the precoded system low average distortion codebooks are derived based on the optimal codeword selection criterion the resulting design is found to relate to the famous applied mathematics problem of subspace packing in the grassmann manifold codebooks designed by this method are proven to provide full diversity order in rayleigh fading channels monte carlo simulations show that limited feedback precoding performs better than antenna subset selection

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

the dual orlicz brunn minkowski theory this paper introduces the dual orlicz brunn minkowski theory for star sets a radial orlicz addition of two or more star sets is proposed and a corresponding dual orlicz brunn minkowski inequality is established based on a radial orlicz linear combination of two star sets a formula for the dual orlicz mixed volume is derived and a corresponding dual orlicz minkowski inequality proved the inequalities proved yield as special cases the precise duals of the conjectured log brunn minkowski and log minkowski inequalities of boroczky lutwak yang and zhang a new addition of star sets called radial m addition is also introduced and shown to relate to the radial orlicz addition

reconstruction of tomographic images from limited range projections using discrete radon transform and tchebichef moments this paper presents an image reconstruction method for x ray tomography from limited range projections it makes use of the discrete radon transform and a set of discrete orthogonal tchebichef polynomials to define the projection moments and the image moments by establishing the relationship between these two sets of moments we show how to estimate the unknown projections from known projections in order to improve the image reconstruction simulation results are provided in order to validate the method and to compare its performance with some existing algorithms

grassmannian beamforming for multiple input multiple output wireless systems transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple input multiple output mimo wireless systems unfortunately optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector both are hard to realize in this article a quantized maximum signal to noise ratio snr beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter by using the distribution of the optimal beamforming vector in independent and identically distributed rayleigh fading matrix channels the codebook design problem is solved and related to the problem of grassmannian line packing the proposed design criterion is flexible enough to allow for side constraints on the codebook vectors bounds on the codebook size are derived to guarantee full diversity order results on the density of grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or snr loss monte carlo simulations are presented that compare the probability of error for different quantization strategies

design and analysis of mimo spatial multiplexing systems with quantized feedback this paper investigates the problem of transmit beamforming in multiple antenna spatial multiplexing sm systems employing a finite rate feedback channel assuming a fixed number of spatial channels and equal power allocation we propose a new criterion for designing the codebook of beamforming matrices that is based on minimizing an approximation to the capacity loss resulting from the limited rate in the feedback channel using the criterion we develop an iterative design algorithm that converges to a locally optimum codebook under the independent identically distributed channel and high signal to noise ratio snr assumption the effect on channel capacity of the finite bit representation of the beamforming matrix is analyzed central to this analysis is the complex multivariate beta distribution and tractable approximations to the voronoi regions associated with the code points furthermore to compensate for the degradation due to the equal power allocation assumption we propose a multimode sm transmission strategy wherein the number of data streams is determined based on the average snr this approach is shown to allow for effective utilization of the feedback bits resulting in a practical and efficient multiple input multiple output system design

fast algorithms for the multidimensional discrete fourier transform in this paper the prime factor algorithm for the evaluation of a one dimensional discrete fourier transform is generalized to the evaluation of multidimensional discrete fourier transforms defined on arbitrary periodic sampling lattices it is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete fourier transform as a sidelight to the derivation of the algorithm a chinese remainder theorem is derived for integer lattices

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

analysis of error in reconstruction of two dimensional signals from irregularly spaced samples we consider the problem of estimating a band limited two dimensional 2 d signal based on a finite set of irregularly spaced samples we derive the minimum mean squared error estimator of the form of a sum of weighted interpolating functions that are identical in shape but are centered at the irregularly spaced sample points and show that this estimator is identical to the minimum energy band limited interpolator that has been previously obtained by others the relation between the mean squared error and the set of irregularly spaced sampling points is studied using the minimax principle the maximum of the mean squared error over a class of signals is minimized by proper choice of the set of sampling points three criteria for evaluating the performance of the sampling point set are derived from a candidate group of sets of irregularly spaced sampling points these criteria may be used to select that point set which is optimal for recovery over a class of signals

the steiner formula for minkowski valuations a steiner type formula for continuous translation invariant minkowski valuations is established in combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations this new steiner type formula is used to obtain a family of brunn minkowski type inequalities for rigid motion intertwining minkowski valuations

noncoherent mimo communication grassmannian constellations and efficient detection this paper considers the design of both a transmitter and a receiver for noncoherent communication over a frequency flat richly scattered multiple input multiple output mimo channel the design is guided by the fact that at high signal to noise ratios snrs the ergodic capacity of the channel can be achieved by input signals that are isotropically distributed on the compact grassmann manifold the first part of the paper considers the design of grassmannian constellations that mimic the isotropic distribution a subspace perturbation analysis is used to determine an appropriate metric for the distance between grassmannian constellation points and using this metric greedy direct and rotation based techniques for designing constellations are proposed these techniques offer different tradeoffs between the minimum distance of the constellation and the design complexity in addition the rotation based technique results in constellations that have lower storage requirements and admit a natural ldquoquasi set partitioningrdquo binary labeling

the dual brunn minkowski theory for bounded borel sets dual affine quermassintegrals and inequalities abstract this paper develops a significant extension of e lutwak s dual brunn minkowski theory originally applicable only to star shaped sets to the class of bounded borel sets the focus is on expressions and inequalities involving chord power integrals random simplex integrals and dual affine quermassintegrals new inequalities obtained include those of isoperimetric and brunn minkowski type a new generalization of the well known busemann intersection inequality is also proved particular attention is given to precise equality conditions which require results stating that a bounded borel set almost all of whose sections of a fixed dimension are essentially convex is itself essentially convex

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

the number of digital straight lines on an n n grid the number of digital straight lines on an n n grid is shown a digital straight line is equivalent to a linear dichotomy of points on a square grid the result is obtained by determining a way of counting the number of linearly separable dichotomies of points on the plane that are not necessarily in general position the analysis is easily modified to provide a simple solution to a similar problem considered by c berenstein and d lavine 1988 on the number of digital straight lines from a fixed starting point

beamforming codebooks for two transmit antenna systems based on optimum grassmannian packings precoding codebook design for limited feedback mimo systems is known to reduce to a discretization problem on a grassmann manifold the case of two antenna beamforming is special in that it is equivalent to quantizing the real sphere the isometry between the grassmannian g2 1 and the real sphere s2 shows that discretization problems in the grassmannian g2 1 are directly solved by corresponding spherical codes notably the grassmannian line packing problem in 2 namely maximizing the minimum distance is equivalent to the tammes problem on the real sphere so that optimum spherical packings give optimum grassmannian packings moreover a simple isomorphism between g2 1 and s2 enables to analytically derive simple codebooks in closed form having low implementation complexity using the simple geometry of some of these codebooks we derive closed form expressions of the probability density function of the relative snr loss due to limited feedback we also investigate codebooks based on other spherical arrangements such as solutions maximizing the harmonic mean of the mutual distances among the codewords which is known as the thomson problem we find that in some special cases grassmannian codebooks based on these other spherical arrangements outperform codebooks from grassmannian packing

sphere packing bounds in the grassmann and stiefel manifolds applying the riemann geometric machinery of volume estimates in terms of curvature bounds for the minimal distance of packings codes in the grassmann and stiefel manifolds will be derived and analyzed in the context of space time block codes this leads to a monotonically increasing minimal distance lower bound as a function of the block length this advocates large block lengths for the code design

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

differential feedback of geometrical mean decomposition precoder for time correlated mimo systems for a time correlated channel we consider the differential feedback of the geometrical mean decomposition gmd precoder which is known to be optimal for a number of criteria when the channel varies slowly we can expect the optimal gmd precoders of consecutive channel uses to be close we consider the feedback of the so called differential precoder and show that it lies in a neighborhood of the identity matrix using matrix perturbation theory furthermore the radius of the neighborhood is shown to be proportional to a time correlation parameter such a characterization allows for subsequent efficient quantization of the differential precoder we show how to design codewords for the prescribed radius by perturbing the identity matrix and applying qr decomposition on the perturbed matrix simulations are given to demonstrate that with a small feedback rate the performance of the proposed differential gmd comes close to the case when perfect channel state information is available to the transmitter for slowly time varying channels like those in indoor and microcellular environments

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

local properties of intertwining operators on the sphere blaschke s original question regarding the local determination of zonoids or projection bodies has been the subject of much research over the years in recent times this research has been extended to include intersection bodies and it has been shown that neither zonoids nor intersection bodies have local characterizations however it has also been proved that both these classes of bodies admit equatorial characterizations in odd dimensions but not in even dimensions the proofs of these results were mostly analytic using properties of associated spherical integral transforms the cosine transform and the radon transform here we elaborate a general principle showing that such local or equatorial characterization problems are equivalent to corresponding support properties of the spherical operators we discuss this within a general framework for intertwining operators on c functions and apply the results to further geometric constructions namely to certain mean section bodies to lq centroid bodies and to k intersection bodies

extremum problems for the cone volume functional of convex polytopes abstract lutwak yang and zhang defined the cone volume functional u over convex polytopes in r n containing the origin in their interiors and conjectured that the greatest lower bound on the ratio of this centro affine invariant u to volume v is attained by parallelotopes in this paper we give affirmative answers to the conjecture in r 2 and r 3 some new sharp inequalities characterizing parallelotopes in r n are established moreover a simplified proof for the conjecture restricted to the class of origin symmetric convex polytopes in r n is provided