duality between channel capacity and rate distortion with two sided state information we show that the duality between channel capacity and data compression is retained when state information is available to the sender to the receiver to both or to neither we present a unified theory for eight special cases of channel capacity and rate distortion with state information which also extends existing results to arbitrary pairs of independent and identically distributed i i d correlated state information s sub 1 s sub 2 available at the sender and at the receiver respectively in particular the resulting general formula for channel capacity c max sub p u x s sub 1 i u s sub 2 y i u s sub 1 assumes the same form as the generalized wyner ziv 1976 rate distortion function r d min sub p u x s sub 1 p x spl i cap u s sub 2 i u s sub 1 x 1 u s sub 2

on the power of finite automata with both nondeterministic and probabilistic states we study finite automata with both nondeterministic and random states npfa s we restrict our attention to those npfa s that accept their languages with a small probability of error and run in polynomial expected time equivalently we study arthur merlin games where arthur is limited to polynomial time and constant space dwork and stockmeyer siam j comput 19 1990 pp 1011 1023 asked whether these npfa s accept only the regular languages this was known if the automaton has only randomness or only nondeterminism we show that the answer is yes in the case of npfa s with a 1 way input head we also show that if l is a nonregular language then either l or bar l is not accepted by any npfa with a 2 way input head toward this end we define a new measure of the complexity of a language l called its 1 tiling complexity for each n this is the number of tiles needed to cover the 1 s in the characteristic matrix of l namely the binary matrix with a row and column for each string of length le n where entry x y 1 if and only if the string xy in l we show that a language has constant 1 tiling complexity if and only if it is regular from which the result on 1 way input follows our main result regarding the general 2 way input tape follows by contrasting two bounds an upper bound of polylog n on the 1 tiling complexity of every language computed by our model and a lower bound stating that the 1 tiling complexity of a nonregular language or its complement exceeds a function in 2 omega sqrt log n infinitely often the last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n this is our main technical result and its proof extends techniques of frobenius and iohvidov developed for hankel matrices sitzungsber der konigl preuss akad der wiss 1894 pp 407 431 hankel and toeplitz matrices and forms algebraic theory birkhauser boston 1982

a double look at duality the two forms of duality that are encountered most frequently by the systems theorist linear duality and convex duality are examined linear duality has a strong algebraic characterization which extends to other structures such as groups and modules convex duality on the other hand capitalizes so strongly on the vector space structure that the resulting powerful theory which is typically interpreted geometrically loses the algebraic flavor of its roots an algebraic characterization of convex duality is presented that generalizes the standard algebraic characterization of linear duality this provides a link between the two forms of duality most important for the systems theorist the algebraic and geometric interpretations together give a double view of duality as used in systems theory

isoperimetric and analytic inequalities for log concave probability measures we discuss an approach based on the brunn minkowski inequality to isoperimetric and analytic inequalities for probability measures on euclidean space with logarithmically concave densities in particular we show that such measures have positive isoperimetric constants in the sense of cheeger and thus always share poincare type inequalities we then describe those log concave measures which satisfy isoperimetric inequalities of gaussian type the results are precised in dimension 1

dimension raising homomorphisms between lattices of convex bodies abstract we settle the first unsolved case of a problem of p m gruber asked by him in 1991 in 9 namely to investigate the homomorphisms from the lattice of convex bodies of e c to the lattice of convex bodies of e d for c d after establishing dimension bounds for such homomorphism in general we completely describe these homomorphisms for the case d c 1 for c 3 the case d c was settled in 9 the obtained result is then applied to characterise homomorphisms and anti homomorphisms from lattices of convex bodies to lattices of convex functions the relationship between such homomorphisms and anti homomorphisms is furnished by the legendre fenchel transform

computation of total capacity for discrete memoryless multiple access channels the arimoto blahut 1972 algorithm is generalized for computation of the total capacity of discrete memoryless multiple access channels mac in addition a class of mac is defined with the property that the uniform distribution achieves the total capacity these results are based on the specialization of the kuhn tucker condition for the total capacity of the mac and an extension of a known symmetry property for single user channels

the concept of duality for measure projections of convex bodies abstract we show that an involution t on some class of functions on r n which reverses order meaning that if f g then t f t g has often a very specific form actually essentially unique it is done in this paper for the class of s concave functions for which this unique formula is derived these functions are for integer s exactly marginals of convex bodies of dimension n s this understanding is also extended and discussed for other classes of functions and represents from our point of view the abstract description of the concept of duality

optimization of amplify and forward multicarrier two hop transmission in this paper frequency domain relay processing in a two hop transmission system is investigated the relay is constrained to be non regenerative that is the relay is only allowed to perform a symbol by symbol memoryless transformation of its received signals multicarrier modulation e g orthogonal frequency division multiplexing ofdm is utilized to convert each hop into a collection of non interfering parallel subcarriers in contrast to conventional scalar amplify and forward af relays that scale all the subcarriers uniformly it is possible to suppress relay noise and to exploit frequency domain diversity by optimizing the relay scaling coefficients of different subcarriers jointly with subcarrier power allocation at the source transmitter this type of scheme is denoted by multicarrier amplify and forward mcaf although the end to end achievable rate of mcaf is a non concave function of the power allocation vectors its optimization is accomplished with an algorithm o mcaf whose computational complexity grows only quadratically with the number of subcarriers by utilizing a structural property of the problem further motivated by the problem structure a suboptimal algorithm wf mcaf with a linear complexity is also proposed in which each hop performs waterfilling separately over a selected subset of subcarriers for hops with a frequency flat channel response the maximum achievable rate is explicitly derived from the associated optimization for hops with rayleigh fading frequency domain channel responses numerical results are presented and it is illustrated that the proposed low complexity wf mcaf algorithm usually achieves near optimal performance

fixed points of polarity type operators abstract a well known result says that the euclidean unit ball is the unique fixed point of the polarity operator this result implies that if in r n the unit ball of some norm is equal to the unit ball of the dual norm then the norm must be euclidean motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators we consider in a real hilbert space setting a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator we show that if the linear operator is positive definite then the considered equation is uniquely solvable by an ellipsoid otherwise the equation can have several possibly infinitely many solutions or no solution at all our analysis yields a few by products of possible independent interest among them results related to coercive bilinear forms essentially a quantitative convex analytic converse to the celebrated lax milgram theorem from partial differential equations and a characterization of real hilbertian spaces

the concept of duality in convex analysis and the characterization of the legendre transform in the main theorem of this paper we show that any involution on the class of lower semi continuous convex functions which is order reversing must be up to linear terms the well known legendre transform

rogers shephard type inequalities for sections abstract in this paper we address the following question given a measure on r n does there exist a constant c 0 such that for any m dimensional subspace h r n and any convex body k r n the following sectional rogers shephard type inequality holds k k h c sup y r n y k h we show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant c n m n m m we also prove marginal inequalities of the rogers shephard type for 1 s concave when 0 s and logarithmically concave functions

zone diagrams in euclidean spaces and in other normed spaces zone diagrams are a variation on the classical concept of voronoi diagrams given n sites in a metric space that compete for territory the zone diagram is an equilibrium state in the competition formally it is defined as a fixed point of a certain dominance map asano matousek and tokuyama proved the existence and uniqueness of a zone diagram for point sites in the euclidean plane and reem and reich showed existence for two arbitrary sites in an arbitrary metric space we establish existence and uniqueness for n disjoint compact sites in a euclidean space of arbitrary finite dimension and more generally in a finite dimensional normed space with a smooth and rotund norm the proof is considerably simpler than that of asano et al we also provide an example of non uniqueness for a norm that is rotund but not smooth finally we prove existence and uniqueness for two point sites in the plane with a smooth but not necessarily rotund norm

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

characterizing the derivative and the entropy function by the leibniz rule consider an operator t c1 r c r satisfying the leibniz rule functional equation t f g tf g f tg f g c1 r we prove that all solution operators t have the form tf x c x f x d x f x ln f x f c1 r x r where c d c r are suitable continuous functions if t acts on the smaller space ck r for some k 2 there are no further solutions if t maps all of c r into c r c 0 and we only have the entropy function cfln f solution we also consider the case of c1 functions f rn r more generally if t c1 r c r and a1 a2 c r c r are operators satisfying the generalized leibniz rule equation t f g tf a1g a2f tg f g c1 r and some weak additional assumptions the operators a1 and a2 are of a very restricted type and any corresponding solution t has the form tf x c x f x f x p x sgn f x d x ln f x f x p x 1 sgnf x here c d p c1 r are continuous functions with im p 0 and the factor sgnf x may be present or not yielding two different solutions if c 0 a1 and a2 must be equal and are uniquely determined by t a1f x a2f x f x p x 1 sgnf x in the case that c x 0 we show that there are two further types of solutions of the functional equation depending only on x and f x

reverse brunn minkowski and reverse entropy power inequalities for convex measures we develop a reverse entropy power inequality for convex measures which may be seen as an anegeometric inverse of the entropy power inequality of shannon and stam the specialization of this inequality to log concave measures may be seen as a version of milman s reverse brunn minkowski inequality the proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies and on a study of eective supports of convex measures both of which are of independent interest as well as on milman s deep technology of m ellipsoids and on certain information theoretic inequalities as a by product we also give a

the chain rule as a functional equation we consider operators t from c1 r to c r satisfying the chain rule t f g tf g tg f g c1 r and study under which conditions this functional equation admits only the derivative or its powers as solutions we also consider t operating on other domains like ck r for k n0 or k and study the more general equation t f g tf g ag f g c1 r where both t and a map c1 r to c r

volume polar volume and euler characteristic for convex functions abstract functional analogs of the euler characteristic and volume together with a new analog of the polar volume are characterized as non negative continuous sl n and translation invariant valuations on the space of finite convex and coercive functions on r n

a functional equation characterizing the second derivative consider an operator t c2 r c r and isotropic maps a1 a2 c1 r c r such that the functional equation t f g tf g a1g a2f g tg f g c2 r is satisfied on c2 r the equation models the chain rule for the second derivative in which case a1g g 2 and a2f f we show under mild non degeneracy conditions which imply that a1 and a2 are very different from t that a1 and a2 must be of the very restricted form a1f f a2f a2f f p or sgn f f p with p 1 and that any solution operator t has the form tf x ca2 f x f x f x h f x f x h x a2 f x x r for some constant c r and some continuous function h conversely any such map t satisfies the functional equation under some natural normalization condition the only solution of the functional equation is tf f which means that the composition rule with some normalization condition characterizes the second derivative if c 0 t does not depend on the second derivative in this case there are further solutions of the functional equation which we determine too

on capacity of line networks we consider communication through a cascade of discrete memoryless channels dmcs the source and destination node of this cascade are allowed to use coding schemes of arbitrary complexity but the intermediate relay nodes are restricted to process only blocks of a fixed length we investigate how the processing at the relays must be chosen in order to maximize the capacity of the cascade that is the maximum achievable end to end rate between the source and the destination for infinite cascades with fixed intermediate processing length at the relays we prove that this intermediate processing can be chosen to be identical without loss of optimality and that the capacity of the cascade coincides with the rate of the best zero error code of length equal to the block length of the intermediate processing we further show that for fixed and identical intermediate processing at all relays convergence of capacity as the length of the cascade goes to infinity is exponentially fast finally we characterize how the block length of the intermediate processing must scale with the length of the cascade to guarantee a constant end to end rate we prove that it is sufficient that the block length scales logarithmically with the network length in order to achieve any rate above the zero error capacity we show that in many cases of interest logarithmic growth is also necessary

a characterization of the support map abstract in this short note we give a characterization of the support map from classical convexity we show it is the unique additive transformation from the class of closed convex sets in r n which include 0 to the class of positive 1 homogeneous functions on r n this will be a consequence of a theorem about transforms from the class of convex sets to itself which preserve minkowski addition

weak e nets and interval chains we construct weak e nets of almost linear size forcertain types of point sets specifically for planar point sets inconvex position we construct weak 1 r nets of size o r r where r denotes the inverse ackermann function for pointsets along the moment curve in d we constructweak 1 r nets of size r 2poly r where the degree of the polynomial in the exponent depends quadratically on d our constructions result from a reduction to a new problem which we call stabbing interval chains with j tuples given therange of integers n 1 n an interval chain of length k is asequence of k consecutive disjoint nonempty intervals containedin n a j tuple bar p p1 pj is said to stab aninterval chain c i1 ik if eachpi falls on a different interval of c the problem is toconstruct a small size family z of j tuples that stabs allk interval chains in n let z j k n denote the minimum size ofsuch a family z we derive almost tight upper and lower bounds forz j k n for every fixed j our boundsinvolve functions m n of the inverse ackermannhierarchy specifically we show that for j 3 we havez 3 k n n lfloor k 2 rfloor n for all k 6 for each j 4 we construct a pair of functionsp j m q j m almost equalasymptotically such that z j p j m n o n m n andz j q j m n n m n

geometric programming duals of channel capacity and rate distortion we show that the lagrange dual problems of the channel capacity problem with input cost and the rate distortion problem are simple geometric programs upper bounds on channel capacity and lower bounds on rate distortion can be efficiently generated from their duals for channel capacity the geometric programming dual characterization is shown to be equivalent to the minmax kullback leibler kl characterization in csiszar et al 1981 for rate distortion the geometric programming dual is extended to rate distortion with two sided state information a duality by mapping is then given between the lagrange dual problems of channel capacity with input cost and rate distortion which resolves several apparent asymmetries between their primal problems in the familiar form of mutual information optimization problems both the primal and dual problems can be interpreted in a common framework of free energy optimization from statistical physics

polynomial time algorithms for multicast network code construction the famous max flow min cut theorem states that a source node s can send information through a network v e to a sink node t at a rate determined by the min cut separating s and t recently it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re encode the information they receive we demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed we give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity we extend these algorithms to integer capacities and to codes that are tolerant to edge failures

the distance trisector curve abstract given points p and q in the plane we are interested in separating them by two curves c 1 and c 2 such that every point of c 1 has equal distance to p and to c 2 and every point of c 2 has equal distance to c 1 and to q we show by elementary geometric means that such c 1 and c 2 exist and are unique moreover for p 0 1 and q 0 1 c 1 is the graph of a function f r r c 2 is the graph of f and f is convex and analytic i e given by a convergent power series at a neighborhood of every point we conjecture that f is not expressible by elementary functions and in particular not algebraic we provide an algorithm that given x r and e 0 computes an approximation to f x with error at most e in time polynomial in log 1 x e the separation of two points by two trisector curves considered here is a special two point case of a new kind of voronoi diagram which we call the zone diagram and which we investigate in a companion paper

a limit theorem for sets of stochastic matrices abstract the following fact about row stochastic matrices is an easy consequence of well known results for each positive integer n 1 there is a positive integer q q n with the property that if a is any n n stochastic matrix then the sequence of matrices aq a2q a3q converges we prove a generalization of this for sets of stochastic matrices under the hausdorff metric let d be any metric inducing the standard topology on the set of n n real matrices for a matrix a and set of matrices b define d a b to be the infimum of d a b over all b b for two sets of matrices a and b define d a b to be the supremum of d a b over all a a and define d a b to be the maximum of d a b and d b a this is the hausdorff metric on the set of subsets of n n stochastic matrices if a is a set of stochastic matrices and k is a positive integer define a k to be the set of all matrices expressible as a product of a sequence of k matrices from a we prove for each positive integer n there is a positive integer p p n such that if a is any subset of n n stochastic matrices then the sequence of subsets a p a 2p a 3p converges with respect to the hausdorff metric

information flow decomposition for network coding we propose a method to identify structural properties of multicast network configurations by decomposing networks into regions through which the same information flows this decomposition allows us to show that very different networks are equivalent from a coding point of view and offers a means to identify such equivalence classes it also allows us to divide the network coding problem into two almost independent tasks one of graph theory and the other of classical channel coding theory this approach to network coding enables us to derive the smallest code alphabet size sufficient to code any network configuration with two sources as a function of the number of receivers in the network but perhaps the most significant strength of our approach concerns future network coding practice namely we propose deterministic algorithms to specify the coding operations at network nodes without the knowledge of the overall network topology such decentralized designs facilitate the construction of codes that can easily accommodate future changes in the network e g addition of receivers and loss of links

functional brunn minkowski inequalities induced by polarity abstract we prove a new family of inequalities which compare the integral of a geometric convolution of non negative functions with the integrals of the original functions for classical inf convolution this type of inequality is called the prekopa leindler inequality which restricted to indicators of convex bodies gives the classical brunn minkowski inequality the convolution we consider is a different one which arises from the study of the polarity transform for functions while inf convolution arises as the pull back of usual addition of convex functions under the legendre transform our geometric inf convolution arises as the pull back of the second order reversing transform on geometric convex functions called either polarity transform or a transform these are up to linear terms the only order reversing isomorphisms on the class of geometric convex functions we prove that the integral of this new geometric convolution of two functions is bounded from below by the harmonic average of the individual integrals our inequality implies the brunn minkowski inequality as well as some other new inequalities for volumes of bodies our inequalities are intimately connected with busemann s convexity theorem a new variant of which we prove for 1 convex hulls and log concave densities

lyz ellipsoid and petty projection body for log concave functions abstract the aims of this paper are to develop the lyz ellipsoid and petty projection body for log concave functions which correspond to the lyz ellipsoid and petty projection body for convex bodies when restricted to the subclass of characteristic functions moreover the continuous sl n contravariant valuation on a subclass of log concave functions is classified

memoryless multiple access channel with asymmetric noisy state information at the encoders the problem of reliable communication over the memoryless state dependent multiple access channel mac is considered where the encoders and the decoder are provided with various degrees of asymmetric noisy channel state information csi for the case where the encoders observe causal asymmetric noisy csi and the decoder observes complete csi inner and outer bounds to the capacity region which are tight for the sum rate capacity are provided next single letter characterizations for the channel capacity regions under each of the following system settings are established 1 the csi at the encoders are asymmetric deterministic functions of the csi at the decoder and the encoders have noncausal noisy csi 2 the encoders observe asymmetric noisy csi with asymmetric delays and the decoder observes complete csi 3 a degraded message set scenario with asymmetric noisy csi at the encoders and complete and or noisy csi at the decoder the main component in these results is a generalization of a recently introduced converse coding approach for the mac with asymmetric quantized csi at the encoders and herein considerably extended and adapted for the noisy csi setup

convex bodies and norms associated to convex measures isotropy like properties are considered for finite measures with heavy tails as a basic tool we extend k ball s relationship between convex bodies and finite logarithmically concave measures to a larger class of distributions satisfying convexity conditions of the brunn minkowski type

a characterization of duality through section projection correspondence in the finite dimensional setting in this paper we show that the well known duality operation in the context of convex bodies in rn is completely characterized by its property of interchanging sections with projections our results are compared to results by boroczky schneider and artstein milman who showed that in many cases the property of order reversing is sufficient to determine a duality operation up to obvious linear modifications in fact we provide another result that recovers a known characterization of duality by the property of order reversing and up to a mild condition also a characterization of duality by interchanging sections by projections

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

operator equations and domain dependence the case of the schwarzian derivative let k nk n consider maps t ck r c r t ck r c r and a1 a2 ck 1 r c r a1 a2 ck 1 r c r satisfying the operator equation t f g tf g a1g a2f g tgt f g tf g a1g a2f g tg for all f g ck r f g ck r we determine the form of all solutions t a1 a2 t a1 a2 of this equation and study their dependence on the domain of t for k 2k 2 the equation models the second derivative chain rule and the solutions t a1a1 and a2a2 are known t a1a1 and a2a2 are closely related local operators we consider the case k 3k 3 and show that variants of the schwarzian derivative appear in t if t depends non trivially on the third derivative there are d 0d 0 p 2p 2 and h c r h c r such that tf d f f p 1 32 f 2f p 2 f p 2h f f p sgnf a1f f 2a2f a2f f p sgnf the term sgnf may be present or not for k 4k 4 there are no solutions t depending non trivially on f k f k the natural domains for t turn out to be cl r cl r for l 1 2 3 l 1 2 3 and c1 r c1 r for a1a1 and a2a2 if t is restricted to ck r ck r functions with non vanishing derivative we may allow p 0p 0 for p 0p 0 the main term in t is the schwarzian derivative s sf f f 32 f f 2

on the limits of communication with low precision analog to digital conversion at the receiver as communication systems scale up in speed and bandwidth the cost and power consumption of high precision e g 8 12 bits analog to digital conversion adc becomes the limiting factor in modern transceiver architectures based on digital signal processing in this work we explore the impact of lowering the precision of the adc on the performance of the communication link specifically we evaluate the communication limits imposed by low precision adc e g 1 3 bits for transmission over the real discrete time additive white gaussian noise awgn channel under an average power constraint on the input for an adc with k quantization bins i e a precision of log2 k bits we show that the input distribution need not have any more than k 1 mass points to achieve the channel capacity for 2 bin 1 bit symmetric quantization this result is tightened to show that binary antipodal signaling is optimum for any signal to noise ratio snr for multi bit quantization a dual formulation of the channel capacity problem is used to obtain tight upper bounds on the capacity the cutting plane algorithm is employed to compute the capacity numerically and the results obtained are used to make the following encouraging observations a up to a moderately high snr of 20 db 2 3 bit quantization results in only 10 20 reduction of spectral efficiency compared to unquantized observations b standard equiprobable pulse amplitude modulated input with quantizer thresholds set to implement maximum likelihood hard decisions is asymptotically optimum at high snr and works well at low to moderate snrs as well

some functional forms of blaschke santalo inequality we establish new functional versions of the blaschke santalo inequality on the volume product of a convex body which generalize to the non symmetric setting an inequality of ball isometric problems in l p and sections of convex sets phd dissertation cambridge 1986 and we give a simple proof of the case of equality as a corollary we get some inequalities for log concave functions and legendre transforms which extend the recent result of artstein et al mathematika 51 33 48 2004 with its equality case

support functions and mean width for α concave functions abstract in this paper we extend some notions previously defined for log concave functions to the larger domain of the so called concave functions we begin with a detailed discussion of support functions first for log concave functions and then for general concave functions we continue by defining mean width and proving some basic results such as an urysohn type inequality finally we demonstrate how such geometric results can imply poincare type inequalities

order preserving and order reversing operators on the class of convex functions in banach spaces abstract a remarkable result by s artstein avidan and v milman states that up to pre composition with affine operators addition of affine functionals and multiplication by positive scalars the only fully order preserving mapping acting on the class of lower semicontinuous proper convex functions defined on r n is the identity operator and the only fully order reversing one acting on the same set is the fenchel conjugation here fully order preserving reversing mappings are understood to be those which preserve reverse the pointwise order among convex functions are invertible and such that their inverses also preserve reverse such order in this paper we establish a suitable extension of these results to order preserving and order reversing operators acting on the class of lower semicontinuous proper convex functions defined on arbitrary infinite dimensional banach spaces

zone diagrams existence uniqueness and algorithmic challenge a zone diagram is a new variation of the classical notion of the voronoi diagram given points sites mathbf p 1 ldots mathbf p n in the plane each mathbf p i is assigned a region r i but in contrast to the ordinary voronoi diagrams the union of the r i has a nonempty complement the neutral zone the defining property is that each r i consists of all mathbf x in mathbb r 2 that lie closer nonstrictly to mathbf p i than to the union of all the other r j j ne i thus the zone diagram is defined implicitly by a fixed point property and neither its existence nor its uniqueness seem obvious we establish existence using a general fixed point result a consequence of schauder s theorem or kakutani s theorem this proof should generalize easily to related settings say higher dimensions then we prove uniqueness of the zone diagram as well as convergence of a natural iterative algorithm for computing it by a geometric argument which also relies on a result for the case of two sites in an earlier paper many challenging questions remain open

network coding in node constrained line and star networks line and star networks with both node and edge constraints are studied in the network coding framework for line networks the capacity region of the general multiple multicast problem is established the coding theorem is based on a binary linear coding scheme while the converse requires new upper bounds that improve on standard cut based bounds for star networks the multiple unicast problem is examined capacity upper bounds are derived and a simple linear coding scheme is proposed which is based on the combinatorial optimization problem of cycle packing in directed graphs the optimality of this scheme is established for a broad class of demands the connection of node constrained network coding in star networks and index coding with side information is discussed and used to partially characterize the optimal linear code for general rates

order automorphisms of the set of bounded observables let h be a complex hilbert space and denote by bs h the set of all self adjoint bounded linear operators on h in this article we describe the form of all bijective maps no linearity or continuity is assumed on bs h which preserve the order in both directions

volume of the polar of random sets and shadow systems we obtain optimal inequalities for the volume of the polar of random sets generated for instance by the convex hull of independent random vectors in euclidean space extremizers are given by random vectors uniformly distributed in euclidean balls this provides a random extension of the blaschke santalo inequality which in turn can be derived by the law of large numbers the method involves shadow systems their connection to busemann type inequalities and how they interact with functional rearrangement inequalities

the entropy per coordinate of a random vector is highly constrained under convexity conditions the entropy per coordinate in a log concave random vector of any dimension with given density at the mode is shown to have a range of just 1 uniform distributions on convex bodies are at the lower end of this range the distribution with i i d exponentially distributed coordinates is at the upper end and the normal is exactly in the middle thus in terms of the amount of randomness as measured by entropy per coordinate any log concave random vector of any dimension contains randomness that differs from that in the normal random variable with the same maximal density value by at most 1 2 as applications we obtain an information theoretic formulation of the famous hyperplane conjecture in convex geometry entropy bounds for certain infinitely divisible distributions and quantitative estimates for the behavior of the density at the mode on convolution more generally one may consider so called convex or hyperbolic probability measures on euclidean spaces we give new constraints on entropy per coordinate for this class of measures which generalize our results under the log concavity assumption expose the extremal role of multivariate pareto type distributions and give some applications