sample complexity of solving non cooperative games this paper studies the complexity of solving two classes of non cooperative games in a distributed manner in which the players communicate with a set of system nodes over noisy communication channels the complexity of solving each game class is defined as the minimum number of iterations required to find a nash equilibrium ne of any game in that class with epsilon accuracy first we consider the class mathcal g of all n player non cooperative games with a continuous action space that admit at least one ne using information theoretic inequalities a lower bound on the complexity of solving mathcal g is derived which depends on the kolmogorov 2 epsilon capacity of the constraint set and the total capacity of the communication channels our results indicate that the game class mathcal g can be solved at most exponentially fast we next consider the class of all n player non cooperative games with at least one ne such that the players utility functions satisfy a certain differential constraint we derive lower bounds on the complexity of solving this game class under both gaussian and non gaussian noise models finally we derive upper and lower bounds on the sample complexity of a class of quadratic games it is shown that the complexity of solving this game class scales according to theta left frac 1 epsilon 2 right where epsilon is the accuracy parameter

brief paper stabilization of some fractional delay systems of neutral type in this note we give new stability tests which enable one to fully characterize the h stability of systems with transfer function g s r s p s q s e s h where h 0 and p q r are real polynomials in the variable s m for 0 p s q s 1

asymptotic formulas for the dual radon transform and applications caracterisation de la transformation de radon duale d une fonction indefiniment differentiable formules asymptotiques et applications

pid controller design for fractional order systems with time delays abstract classical proper pid controllers are designed for linear time invariant plants whose transfer functions are rational functions of s where 0 1 and s is the laplace transform variable effect of input output time delay on the range of allowable controller parameters is investigated the allowable pid controller parameters are determined from a small gain type of argument used earlier for finite dimensional plants

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

pointwise rates of convergence for the oliker prussner method for the monge ampere equation we study the oliker prussner method exploiting its geometric nature we derive discrete stability and continuous dependence estimates in the max norm by using a discrete alexandroff estimate and the brunn minkowski inequality we show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain nodes still conform to the domain boundary this gives a suitable notion of operator consistency which combined with stability leads to pointwise rates of convergence for classical and non classical solutions of the monge ampere equation

reverse faber krahn and mahler inequalities for the cheeger constant we prove a reverse faber krahn inequality for the cheeger constant stating that every convex body in 2 has an affine image such that the product between its cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle an analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square we also prove a mahler type inequality for the cheeger constant stating that every axisymmetric convex body in 2 has a linear image such that the product between its cheeger constant and the cheeger constant of its polar body is not larger than the same quantity for the square

how to recognize convexity of a set from its marginals abstract we investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter we prove in particular that if a set of finite perimeter has log concave marginals onto a e hyperplane then the set is convex our proof relies on measuring the perimeter of a set through a hilbertian fractional sobolev norm a fact that we believe has its own interest

on the case of equality in the brunn minkowski inequality for capacity suppose that and 1 are convex open subsets of rn denote their convex combination by the brunn minkowski inequality says that vol t 1 t vol 0 1 n t vol for 0 t l moreover if there is equality for some t other than an endpoint then the domains 1 and 0 are translates and dilates of each other borell proved an analogue of the brunn minkowski inequality with capacity defined below in place of volume borel s theorem b says theorem a let t t 1 1 t 0 be a convex combination of two convex subsets of rn n 3 then cap the main purpose of this note is to prove

brunn minkowski inequalities in product metric measure spaces abstract given one metric measure space x satisfying a linear brunn minkowski inequality and a second one y satisfying a brunn minkowski inequality with exponent p 1 we prove that the product x y with the standard product distance and measure satisfies a brunn minkowski inequality of order 1 1 p 1 under mild conditions on the measures and the assumption that the distances are strictly intrinsic the same result holds when we consider restricted classes of sets we also prove that a linear brunn minkowski inequality is obtained in x y when y satisfies a prekopa leindler inequality in particular we show that the classical brunn minkowski inequality holds for any pair of weakly unconditional sets in r n i e those containing the projection of every point in the set onto every coordinate subspace when we consider the standard distance and the product measure of n one dimensional real measures with positively decreasing densities this yields an improvement of the class of sets satisfying the gaussian brunn minkowski inequality furthermore associated isoperimetric inequalities as well as recently obtained brunn minkowski s inequalities are derived from our results

the spectra of uniform hypertrees abstract in this paper we study the spectra of uniform hypertrees by using the generalized weighted incident matrix we show that is a nonzero eigenvalue of the hypertree h corresponding to an eigenvector with all elements nonzero if and only if is a root of the polynomial h i 0 m 1 i m i x m i r where m i is the number of matchings of order i in h

some applications of mass transport to gaussian type inequalities as discovered by brenier mapping through a convex gradient gives the optimal transport in n in the present article this map is used in the setting of gaussian like measures to derive an inequality linking entropy with mass displacement by a straightforward argument as a consequence logarithmic sobolev and transport inequalities are recovered finally a result of caffarelli on the brenier map is used to obtain gaussian correlation inequalities

unimodality convexity and applications properties of univariate unimodal distributions concepts of multivariate unimodality some more notions of unimodality unimodality for discrete distributions unimodality of infinitely divisible distributions unimodality and notions of dependence ordering of distributions by peakedness applications of unimodality in statistical inference convexity in reliability theory appendix

the discrete planar l0 minkowski problem abstract in the discrete setting the l 0 minkowski problem extends the question posed and answered by the classical minkowski s existence theorem for polytopes in particular the planar extension which we address in this paper concerns the existence of a convex polygonal body which contains the origin whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area

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

the minimum excluded volume of convex shapes we prove that the excluded volume of convex shapes and with equal volumes attains its minimum only if coincides with the inversion of relative to its centroid it follows that if is a cylindrically symmetric shape and is a replica of rotated in space then the excluded volume of and attains its minimum when and are in the antiparallel configuration this suggests that ensembles of such hard particles may experience the same geometric frustration as spins in the ising model for antiferromagnetism on certain lattices

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

brief analysis of fractional delay systems of retarded and neutral type this paper analyses several properties linked to the robust control of fractional differential systems with delays the bibo stability of both retarded and neutral fractional delay systems is related to the location of their poles in the particular case of retarded systems we give some properties of the poles of the system and the singular values of its hankel operator

shadows of convex bodies it is proved that if c is a convex body in rn then c has an affine image c of nonzero volume so that if p is any 1 codimensional orthogonal projection pca 1 cl n l n it is also shown that there is a pathological body k all of whose orthogonal projections have volume about vx times as large as iki n 1 n 0 introduction the problems discussed in this paper concern the areas of shadows orthogonal projections of convex bodies and to a lesser extent the surface areas of such bodies if c is a convex body in irn and 0 a unit vector p c will denote the orthogonal projection of c onto the 1 codimensional space perpendicular to 0 volumes and areas of convex bodies and their surfaces will be denoted with i 1 the relationship between shadows and surface areas of convex bodies is expressed in cauchy s well known formula for each n e n let vn be the volume of the n dimensional euclidean unit ball and let a an be the rotationally invariant probability on the unit sphere sn cauchy s formula states that if c is a convex body in rn then its surface area is ac i s p6ci dcr o vni jn i the classical isoperimetric inequality in rn states that any body has surface area at least as large as a euclidean ball of the same volume the first section of this paper is devoted to the proof of a local isoperimetric inequality showing that all bodies have large shadows rather than merely large surface area or average shadow the principal motivation for this result is its relationship to a conjecture of vaaler and the important problems surrounding it this theorem and its connection with vaaler s conjecture are described at the beginning of 1 an important role is played in the theory of convex bodies by the so called projection body of a convex body it is easily seen by considering polytopes received by the editors september 12 1989 1980 mathematics subject classification 1985 revision primary 52a20 52a40 the author was supported in part by nsf dms 8807243 1991 american mathematical society 0002 9947 91 1 00 25 per page

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

the busemann petty problem in hyperbolic and spherical spaces abstract the busemann petty problem asks whether origin symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume it is known that the answer to this problem is affirmative if n 4 and negative if n 5 we study this problem in hyperbolic and spherical spaces

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

the borell ehrhard game a precise description of the convexity of gaussian measures is provided by sharp brunn minkowski type inequalities due to ehrhard and borell we show that these are manifestations of a game theoretic mechanism a minimax variational principle for brownian motion as an application we obtain a gaussian improvement of barthe s reverse brascamp lieb inequality

brunn minkowski and prekopa leindler s inequalities under projection assumptions abstract brunn minkowski s theorem says that vol 1 k l 1 n for k l convex bodies is a concave function in and assuming a common hyperplane projection of k and l it is known that the volume itself is concave the a priori natural hypothesis of a common n k plane projection of the sets turned out in the end not to imply the 1 k th concavity of the volume function in this paper we show which is the somehow best projection type assumption that is needed in order to get concavity for vol 1 k l 1 k characterizing also the equality case in the corresponding inequality moreover we consider the same problem for its functional analogue the prekopa leindler inequality

volume of mixed bodies by using inequalities obtained for the volume of mixed bodies and the petty projection inequality sharp isoperimetric inequalities are derived for the projection measures quermassintegrale of a convex body these projection measure inequalities which involve mixed projection bodies zonoids are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body the inequality obtained for the volume of mixed bodies is also used to derive a form of the brunn minkowski inequality involving mixed bodies as an application inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies o introduction in 24 it was shown that the petty projection inequality 26 will yield extensions of the known isoperimetric inequalities between the volume of a convex body and its projection measures quermassintegrale in this article we show that the petty projection inequality can be used to obtain sharp isoperimetric inequalities corollary 5 20 which are strengthened forms of the classical inequalities between the projection measures of a convex body this is accomplished by using the so called mixed convex bodies introduced by firey 17 geometric inequalities are derived for the volume of mixed bodies theorems 4 1 and 4 3 which when combined with the petty projection inequality yield inequalities involving projection bodies zonoids which are related to the inequalities between the projection measures of a convex body the inequalities for mixed bodies are also used to obtain a version of the brunn minkowski inequality theorem 4 2 involving mixed bodies also derived are inequalities between the volume of convex bodies and their associated mixed projection integral theorems 6 4 and 6 4 that are extensions and improvements of some previously established inequalities since some of our results concern projection bodies a topic of interest to workers outside the area of geometric convexity we have restated a few basic definitions and standard results from the field of geometric inequalities in 1 we collect for quick reference some well known results concerning mixed volumes and mixed area measures which will be required later references for material presented in this section are bonnesen and fenchel 7 for mixed volumes busemann 9 and leichtweiss 20 the original works of aleksandrov 1 4 and fenchel and jessen 15 supplemented by the survey of schneider 28 are recommended references for mixed area measures 2 contains introductory material regarding blaschke addition received by the editors january 8 1985 1980 mcithematics subject classihecition primary 52a40 secondary 53a15 46b20

metric marginal problems for set valued or non measurable variables in a separable metric space if two borel probability measures laws are nearby in a suitable metric then there exist random variables with those laws which are nearby in probability specifically by a well known theorem of strassen the prohorov distance between two laws is the infimum of ky fan distances of random variables with those laws the present paper considers possible extensions of strassen s theorem to two random elements one of which may be compact set valued and or non measurable there are positive results in finite dimensional spaces but with factors depending on the dimension examples show that such factors cannot entirely be avoided so that the extension of strassen s theorem to the present situation fails in infinite dimensions

on a reverse form of the brascamp lieb inequality we prove a reverse form of the multidimensional brascamp lieb inequality our method also gives a new way to derive the brascamp lieb inequality and is rather convenient for the study of equality cases

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

on the number of solutions to the discrete two dimensional l0 minkowski problem abstract our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices if they exist such polygons are solutions to the discrete l 0 minkowski problem in the plane proposed in a larger generality by lutwak in his extension of the brunn minkowski theory the method exploits the equivalence between the l 0 polygons and the homothetic solutions to appropriate anisotropic crystalline flows hence as a by product we obtain a positive answer to jean taylor s conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature without any symmetry assumptions

fischer type determinantal inequalities for accretive dissipative matrices abstract let a a 11 a 12 a 21 a 22 be an n n accretive dissipative matrix k and l be the orders of a 11 and a 22 respectively and let m min k l then det a a det a 11 det a 22 where a 2 3 m 2 if m n 3 2 n 2 if n 3 m n 2 this improves a result of ikramov

generalizations of the brunn minkowski inequality abstract yuan and leng 2007 gave a generalization of the matrix form of the brunn minkowski inequality in this note we first give a simple proof of this inequality and then show a generalization of this to a larger class of matrices namely matrices whose numerical ranges are contained in a sector

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

inequalities for mixed projection bodies mixed projection bodies are related to ordinary projection bodies zonoids in the same way that mixed volumes are related to ordinary volume analogs of the classical inequalities from the brunn minkowski theory such as the minkowski brunn minkowski and aleksandrov fenchel inequalities are developed for projection and mixed projection bodies

radon transforms on grassmann manifolds if n p q 1 the affine grassmann manifolds g p n of p planes and g q n of q planes in rn are dual homogeneous spaces of the euclidean motion group m n with g p n and g q n being incident if and only if meets at a right angle the resulting integral transform from functions on g p n to functions on g q n generalizes both the classical radon transform and its dual we show that this transform intertwines the actions of certain m n invariant differential operators on g p n and g q n and we prove an inversion formula for this transform when n is odd generalizing the radon inversion formula and obtaining in particular an inversion formula for the dual transform

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

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

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

theory of surround system and associated inequalities in this paper we define the surround system s m c l and the surround system with a center s 2 p c l under the proper hypotheses we establish several geometric inequalities for s 2 p c l as applications we give the background of the surround system with a center s 2 p c l in space science

on some generalizations of the brunn minkowski inequality abstract recently liu 2016 9 obtained some determinantal inequalities for sector matrices in this paper we make considerable improvement on liu s results

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

stabilization of miso fractional systems with delays abstract we consider multi input single output miso fractional systems of commensurate fractional orders with different input or output delays we derive explicit expressions of left and right coprime factorizations over h and of the associated bezout factors of the transfer matrix of the systems these factors allow the construction of the youla kucera parametrization of the set of stabilizing controllers which guarantee the internal stability of the closed loop systems

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

capacitary inequalities of the brunn minkowski type the main purpose of this paper is to show how so called a concave capacities may be built up from newton capacity c in lr n 3 in much the same way as concave measures from lebesgue measure m n in lr n 1 throughout o f2 o stand for non empty open and convex subsets of ir and j o denotes the class of all convex bodies contained in f2 recall that a positive radon measure in f2 is said to be concave if a and

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

an algorithm for stabilization of fractional order time delay systems using fractional order pid controllers this technical note presents a solution to the problem of stabilizing a given fractional order system with time delay using fractional order pllambdadmu controllers it is based on determining a set of global stability regions in the kp ki kd space corresponding to the fractional orders lambda and mu in the range of 0 2 and then choosing the biggest global stability region in this set this method can be also used to find the set of stabilizing controllers that guarantees prespecified gain and phase margin requirements the algorithm is simple and has reliable result which is illustrated by an example and hence is practically useful in the analysis and design of fractional order control systems

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

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

cone volume measure and stability abstract we prove stability results for two central inequalities involving the cone volume measure of a centered convex body the subspace concentration conditions and the u functional volume inequality

volumes of restricted minkowski sums and the free analogue of the entropy power inequality in noncommutative probability theory independence can be based on free products instead of tensor products this yields a highly noncommutative theory free probability theory for an introduction see 9 the analogue of entropy in the free context was introduced by the second named author in 8 here we show that shannon s entropy power inequality 6 1 has an analogue for the free entropy x theorem 2 1

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

rate based flow controllers for communication networks in the presence of uncertain time varying multiple time delays an h based robust controller is designed for a rate feedback flow control problem in single bottleneck communication networks the controller guarantees stability robustness to uncertain time varying multiple time delays in different channels it also brings the queue length at the bottleneck node to the desired steady state value asymptotically and satisfies a weighted fairness condition lower bounds for stability margins for uncertainty in the time delays and for the rate of change of the time delays are derived a number of simulations are included to demonstrate the time domain performance of the controller trade offs between robustness and time domain performance are also discussed