volume inequalities for isotropic measures a direct approach to ball s simplex inequality is presented this approach which does not use the brascamp lieb inequality also gives barthe s characterization of the simplex for ball s inequality and extends it from discrete to arbitrary measures it also yields the dual inequality along with equality conditions and it does both for arbitrary measures

h optimal sampled data control in continuous time systems we consider the design of h optimal discrete time digital controllers in continuous time systems an apparent difficulty especially in utilizing modern transform domain analysis in this context stems from the absence of an appropriate transfer function model for the hybrid time discrete and continuous closed loop system this difficulty is overcome through the introduction of an equivalent difference equation model for the continuous time system with distributed inputs and outputs equivalence being in the sense that the continuous and discrete time inputs and outputs are essentially identical using the interplay between the discrete and the continuous time models solutions of the well known purely continuous time and purely discrete time standard problems extend to solutions of the problem at hand they comprise riccati equation characterizations of feasible combinations of sampling rates and bounds on the closed loop induced input output norm and parameterization of compensators we consid

a constructive approach to approximate linear periodic systems in this note the problem on approximating a linear periodic system with period n by a periodic model with period m 0 m n is revisited different from the existing results the proposed method is based on state space representations and the approximation performance is evaluated using hinfin norms under this framework both finite impulse response fir and infinite impulse response iir periodic systems can be efficiently approximated by using a constructive linear matrix inequality approach finally some numerical examples are given to illustrate the effectiveness of the proposed approach

multiple runs distributions recurrences and critical values abstract most traditional applications of runs tests have entailed bivariate runs distributions extensions to more general runs tests for randomness have been hindered by the lack of exact critical values for distributions with more than two kinds of elements combinatorial recurrences that can be used to compute exact significance levels for small sample sizes are presented for runs distributions for 2 kinds of elements critical values for small sample sizes are tabled for 2 1 6 more extensive tables are available from the author

crame spl acute r rao and moment entropy inequalities for renyi entropy and generalized fisher information the moment entropy inequality shows that a continuous random variable with given second moment and maximal shannon entropy must be gaussian stam s inequality shows that a continuous random variable with given fisher information and minimal shannon entropy must also be gaussian the crame spl acute r rao inequality is a direct consequence of these two inequalities in this paper the inequalities above are extended to renyi entropy p sup th moment and generalized fisher information generalized gaussian random densities are introduced and shown to be the extremal densities for the new inequalities an extension of the crame spl acute r rao inequality is derived as a consequence of these moment and fisher information inequalities

the logarithmic minkowski problem in analogy with the classical minkowski problem necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone volume measure of the unit ball of a finite dimensional banach space

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

new affine measures of symmetry for convex bodies abstract grunbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one here we introduce new measures of symmetry that measure how far a given convex body is from one with enough symmetries to define these new measures of symmetry we use affine covariant points we give examples of convex bodies whose affine covariant points are far apart in particular we give an example of a convex body whose centroid and santalo point are far apart

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

distribution of runs and longest runs a new generating function approach exact distributions of run statistics are traditionally obtained using combinatorial methods which under certain situations become very tedious run distributions of multiple object systems although appearing frequently in applications from various fields such as computational biology are not commonly used due in part to the lack of easy to use formulas in this article a method for evaluating partition functions of lattice models in the field of statistical mechanics is used to develop a systematic method to study various run statistics in multiple object systems by using particular generating functions for the specified situation under study many new distributions can be obtained in a unified and coherent way the method makes it possible to manipulate formulas of run statistics by using binomial identities to obtain more general yet simpler formulas to illustrate the applications of the general method the distributions of the total number of runs and the longest runs are investigated nove

performance analysis of sampled data control systems abstract this brief paper considers multivariable sampled data systems with bandlimited exogenous inputs a procedure is given for the computation of the l 2 induced input to output norm of such a system in addition sufficient conditions are given for input output stability the digital implementation of an analog controller will generally degrade the performance of the closed loop system in light of this two examples are given to illustrate how this degradation is quantified as a function of the sampling period in general this allows the determination of the minimum sampling rate required to recover the performance designed for using the original continuous time setup

intersection bodies in r4 intersection bodies were introduced in 1988 by lutwak who found a close connection between those bodies and the well known 1956 busemann petty problem which asks whether origin symmetric convex bodies with larger central hyperplane sections also have greater volume the author has recently shown that an origin symmetric star bodykis an intersection body if and only if the function x 1kis a positive definite distribution where x k min a 0 x ak we use this result to prove that the unit balls of the spaces l4q 2 q are intersection bodies forn 5 the unit balls of the spaces lnq 2 q are not intersection bodies the technique of this paper allows to find precise expressions for the generating measures signed measures of these bodies the result that the unit cube in r4is an intersection body shows that the negative solution to the busemann petty problem forn 4 in 20 was incorrect

constrained convex bodies with extremal affine surface areas abstract given a convex body k r n and p r we introduce and study the extremal inner and outer affine surface areas i s p k sup k k as p k and o s p k inf k k as p k where as p k denotes the l p affine surface area of k and the supremum is taken over all convex subsets of k and the infimum over all convex compact subsets containing k the convex body that realizes i s 1 k in dimension 2 was determined in 3 where it was also shown that this body is the limit shape of lattice polytopes in k in higher dimensions no results are known about the extremal bodies we use a thin shell estimate of 23 and the lowner ellipsoid to give asymptotic estimates on the size of i s p k and o s p k surprisingly it turns out that both quantities are proportional to a power of volume

the cramer rao inequality for star bodies associated with each body k in euclidean n space mathbb r sp n is an ellipsoid gamma sb 2k called the legendre ellipsoid of k it can be defined as the unique ellipsoid centered at the body s center of mass such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of the body in an earlier paper the authors showed that corresponding to each convex body k subset mathbb r sp n is a new ellipsoid gamma sb 2 k that is in some sense dual to the legendre ellipsoid the legendre ellipsoid is an object of the dual brunn minkowski theory while the new ellipsoid gamma sb 2 k is the corresponding object of the brunn minkowski theory the present paper has two aims the first is to show that the domain of gamma sb 2 can be extended to star shaped sets the second is to prove that the following relationship exists between the two ellipsoids if k is a star shaped set then gamma sb 2 k subset gamma sb 2k with equality if and only if k is an ellipsoid centered at the origin this inclusion is the geometric analogue of one of the basic inequalities of information theory the cramer rao inequality

information theoretic inequalities for contoured probability distributions we show that for a special class of probability distributions that we call contoured distributions information theoretic invariants and inequalities are equivalent to geometric invariants and inequalities of bodies in euclidean space associated with the distributions using this we obtain characterizations of contoured distributions with extremal shannon and renyi entropy we also obtain a new reverse information theoretic inequality for contoured distributions

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

fisher information and matrix valued valuations abstract all continuous and affinely contravariant matrix valued valuations on the sobolev space w 1 2 r n are completely classified it is shown that there is a unique such valuation this valuation turns out to be the fisher information matrix

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

decomposition and approximation of periodic systems this paper proposes a decomposition of periodically time varying systems in an attempt to reduce the often overwhelming number of modeling parameters involved in many cases it would be possible to represent the system using only a few of those components an a priori bound on the approximation error is derived in terms of the smoothness of the system parameters both continuous and discrete time systems are considered

on the similarity of the entropy power inequality and the brunn minkowski inequality corresp the entropy power inequality states that the effective variance entropy power of the sum of two independent random variables is greater than the sum of their effective variances the brunn minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii both these inequalities are recast in a form that enhances their similarity in spite of this similarity there is as yet no common proof of the inequalities nevertheless their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found two applications of this reasoning are presented first an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the fisher information matrix given an entropy constraint just as a sphere minimizes the surface area given a volume constraint second a theorem involving the effective radii of growing convex sets is proved

toward a relative q entropy abstract we address the question and related controversy of the formulation of the q entropy and its relative entropy counterpart for models described by continuous non discrete sets of variables we notice that an l p normalized functional proposed by lutwak yang zhang lyz which is essentially a variation of a properly normalized relative renyi entropy up to a logarithm has extremal properties that make it an attractive candidate which can be used to construct such a relative q entropy we comment on the extremizing probability distributions of this lyz functional its relation to the escort distributions a generalized fisher information and the corresponding cramer rao inequality we point out potential physical implications of the lyz entropic functional and of its extremal distributions

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

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

fourier state space analysis of linear discrete time periodic systems abstract by the floquet lyapunov transformation a linear periodically time varying discrete time system has a state space representation a b n c n having time invariant autonomous dynamics while the time periodicity is contained in the input and output matrices in this paper it is shown that such a system also has state space realizations of the form a b n c and a b c n where the time periodicity is contained at the input or alternatively the output matrix only and whose state dimensions are bounded by the product of the system order and the period length the system representations are applied to construct a state space realization of the optimal linear time invariant approximation of a time periodic system

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

moment entropy inequalities for a random vector the pth moment matrix is defined for a real random vector generalizing the classical covariance matrix sharp inequalities relating the pth moment and renyi entropy are established generalizing the classical inequality relating the second moment and the shannon entropy the extremal distributions for these inequalities are completely characterized

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

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 spherical convex floating body abstract for a convex body on the euclidean unit sphere the spherical convex floating body is introduced the asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated this gives rise to a new spherical area measure the floating area remarkably this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry several properties of the floating area are established

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

a steiner formula in the lp brunn minkowski theory abstract we prove an analogue of the classical steiner formula for the l p affine surface area of a minkowski outer parallel body for any real parameter p we show that the classical steiner formula and the steiner formula of lutwak s dual brunn minkowski theory are special cases of this new steiner formula this new steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature they have the intrinsic volumes of the classical brunn minkowski theory as well as the dual quermassintegrals of the dual brunn minkowski theory as special cases properties of these new quantities are investigated a connection to information theory among them a steiner formula for the s th mixed l p affine surface area of a minkowski outer parallel body for any real parameters p and s is also given

positions of convex bodies associated to extremal problems and isotropic measures abstract we show that there are close relations between extremal problems in dual brunn minkowski theory and isotropic type properties for some borel measures on the sphere the methods we use allow us to obtain similar results in the context of firey brunn minkowski theory we also study reverse inequalities for dual mixed volumes which are related with classical positions such as l position or isotropic position

renyi divergence and lp affine surface area for convex bodies abstract we show that the fundamental objects of the l p brunn minkowski theory namely the l p affine surface areas for a convex body are closely related to information theory they are exponentials of renyi divergences of the cone measures of a convex body and its polar we give geometric interpretations for all renyi divergences d not just for the previously treated special case of relative entropy which is the case 1 now no symmetry assumptions are needed and if at all only very weak regularity assumptions are required previously the relative entropies appeared only after performing second order expansions of certain expressions now already first order expansions make them appear thus in the new approach we detect faster details about the boundary of a convex body

on the l p monge ampere equation abstract for the case where 0 p 1 we prove the existence of solutions of the l p monge ampere equation for arbitrary measures and show that the solution is in general not unique

characterization of the cal l _2 induced norm for linear systems with jumps with applications to sampled data systems this paper considers a continuous time linear system with finite jumps at discrete instants of time an iterative method to compute the cal l 2 induced norm of a linear system with jumps is presented each iteration requires solving an algebraic riccati equation it is also shown that a linear feedback interconnection of a continuous time finite dimensional linear time invariant fdlti plant and a discrete time finite dimensional linear shift invariant fdlsi controller can be represented as a linear system with jumps this leads to an iterative method to compute the cal l 2 induced norm of a sampled data system

anisotropic fractional perimeters the anisotropic fractional s perimeter with respect to a convex body k in rn is shown to converge as s 1 to the anisotropic perimeter with respect to the moment body of k for anisotropic fractional s seminorms on bv rn the corresponding result is established generalizing results of bourgain brezis mironescu and davila minimizers of the anisotropic fractional isoperimetric inequality with respect to k are shown to converge to the moment body of k as s 1 anisotropic fractional sobolev inequalities are established for a borel set e r and 0 s 1 the fractional s perimeter of e is given by ps e

lp minkowski problem with not necessarily positive data abstract the l p minkowski problem in the plane has been the object of a number of recent investigations here we present various existence results for 0 p 2 in the case 0 p 2 a necessary and sufficient condition for the solvability of the problem is given

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

divergence for s concave and log concave functions abstract we prove new entropy inequalities for log concave and s concave functions that strengthen and generalize recently established reverse log sobolev and poincare inequalities for such functions this leads naturally to the concept of f divergence and in particular relative entropy for s concave and log concave functions we establish their basic properties among them the affine invariant valuation property applications are given in the theory of convex bodies

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

multinomial runs tests to detect clustering in constrained free recall abstract psychologists often want to detect category structure in subjects free recall protocols although runs tests based on the binomial distribution are commonly used to detect nonrandomness within a sequence many research situations require tests based on the multinomial distribution we propose a test of randomness versus clustering based on the number of runs in multinomial data we illustrate its use with data from a mass communication experiment using a constrained free recall procedure

tensor valuations on lattice polytopes abstract the ehrhart polynomial and the reciprocity theorems by ehrhart macdonald are extended to tensor valuations on lattice polytopes a complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant every such valuation is a linear combination of the ehrhart tensors which is shown to no longer hold true for rank nine

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

intersection bodies positive definite distributions and the busemann petty problem the 1956 busemann petty problem asks whether symmetric convex bodies with larger central hyperplane sections also have greater volume in 1988 lutwak introduced the concept of an intersection body which is closely related to the busemann petty problem we prove that an origin symmetric star body k in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is an intersection body if and only if x 1 k is a positive definite distribution on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i where x k min a 0 x ak we use this result to show that for every dimension n there exist polytopes in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i which are intersection bodies for example the cross polytope the unit ball of every subspace of l p 0 p 2 is an intersection body the unit ball of the space l n q 2 q n 5 using lutwak s connection with the busemann petty problem we present new counterexamples to the problem for n 5 and confirm the conjecture of meyer that the answer to the problem is affirmative if the smaller body is a polar projection body

run probabilities in sequences of markov dependent trials abstract the probability of the occurrence of a run r is obtained as a function of the composition of r the number n of trials and the probabilities of the v 2 possible outcomes at each trial the run r can consist of any specified sequence of outcomes and the probability that one or more of a given collection of runs occurs is also evaluated the probabilities of the v possible outcomes can vary arbitrarily from trial to trial and can be l order markov dependent on the l preceding outcomes the practical application of these results is discussed

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

on runs and longest run tests a method of finite markov chain imbedding abstract in this article under very general assumptions on bernoulli trials a simple numerical method based on the finite markov chain imbedding technique is systematically developed to determine a the joint distribution of the number of success runs and the number of successes b the conditional distribution of the number of success runs given the number of successes that is the exact distribution of the success runs test statistic c the joint distribution of the length of the longest success run and the number of successes and d the conditional distribution of the length of the longest success run given the number of successes that is the exact distribution of the longest success run test statistic the critical regions and powers of these two tests success runs and longest success run are derived under the null hypothesis of independence and identical distribution iid as well as under the alternative hypothesis of markov dependence the application of the success runs test is illust

second derivative test for intersection bodies in 1956 busemann and petty asked whether symmetric convex bodies in rnwith larger central hyperplane sections also have greater volume this question was answered in the negative forn 5 in a series of papers giving individual counterexamples in 1988 lutwak introduced the concept of an intersection body and proved that every smooth nonintersection body in rnprovides a counterexample to the busemann petty problem in this article we use the connection between intersection bodies and positive definite distributions established by the author in an earlier paper to give a necessary condition for intersection bodies in terms of the second derivative of the norm this result allows us to produce a variety of counterexamples to the busemann petty problem in rn n 5 for example the unit ball of theq sum of any finite dimensional normed spacesxandywithq 2 dim x 1 dim y 4 is not an intersection body as well as the unit balls of the orlicz spaces lnm n 5 withm 0 m 0 0

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

relative entropies for convex bodies we introduce a new class of not necessarily convex bodies and show among other things that these bodies provide yet another link between convex geometric analysis and information theory namely they give geometric interpretations of the relative entropy of the cone measures of a convex body and its polar and related quantities such interpretations were first given by paouris and werner for symmetric convex bodies in the context of the lp centroid bodies there the relative entropies appear after performing second order expansions of certain expressions now no symmetry assumptions are needed moreover using the new bodies already first order expansions make the relative entropies appear thus these bodies detect faster details of the boundary of a convex body than the lp centroid bodies

maximum entropy reconstruction using derivative information part 1 fisher information and convex duality maximum entropy spectral density estimation is a technique for reconstructing an unknown density function from some known measurements by maximizing a given measure of entropy of the estimate here we present a variety of new entropy measures which attempt to control derivative values of the densities our models apply among others to the inference problem based on the averaged fisher information measure the duality theory we develop resembles models used in convex optimal control problems we present a variety of examples including relaxed moment matching with fisher information and best interpolation on a strip

new sine ellipsoids and related volume inequalities abstract corresponding to the legendre ellipsoid and the lyz ellipsoid two new sine ellipsoids are introduced in this paper these four ellipsoids are closely related in the pythagorean relation and duality several volume inequalities and the valuation properties are obtained for two new ellipsoids

relative entropy of cone measures and lp centroid bodies let k be a convex body in r n we introduce a new affine invariant which we call k t hat can be found in three different ways a as a limit of normalized lp affine surface areas b as the relative entropy of the cone measure of k and the cone measure of k c as the limit of the volume difference of k and lp centroid bodies we investigate properties of k and of related new invariant quantities in particular we show new affine isoperimetric inequalities and we show an information inequality for convex bodies

the lp minkowski problem for n p 1 abstract chou and wang s existence result for the l p minkowski problem on s n 1 for p n 1 and an absolutely continuous measure is discussed and extended to more general measures in particular we provide an almost optimal sufficient condition for the case p 0 1

affine moments of a random vector an affine invariant pth moment measure is defined for a random vector and used to prove sharp moment entropy inequalities that are more general and stronger than standard moment entropy inequalities

an extension of minkowski s theorem and its applications to questions about projections for measures abstract minkowski s theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body and the surface area measure determines a convex body uniquely up to a shift in this manuscript we prove an extension of minkowski s theorem consider a measure on r n with positive degree of concavity and positive degree of homogeneity we show that a surface area measure of a convex set k weighted with respect to determines a convex body uniquely up to measure zero we also establish an existence result under natural conditions including symmetry we apply this result to extend the solution to classical shephard s problem to do this we introduce a new notion which relates projections of convex bodies to a given measure and is a generalization of the lebesgue volume of a projection

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1