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

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

modeling highway traffic headway distributions using superstatistics we study traffic clearance distributions i e the instantaneous gap between successive vehicles and time headway distributions by applying the beck and cohen superstatistics we model the transition from free phase to congested phase with the increase of vehicle density as a transition from the poisson statistics to that of the random matrix theory we derive an analytic expression for the spacing distributions that interpolates from the poisson distribution and wigner s surmise and apply it to the distributions of the net distance and time gaps among the succeeding cars at different densities of traffic flow the obtained distribution fits the experimental results for single vehicle data of the dutch freeway a9 and the german freeway a5

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

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

superstatistical fluctuations in time series applications to share price dynamics and turbulence we report a general technique to study a given experimental time series with superstatistics crucial for the applicability of the superstatistics concept is the existence of a parameter beta that fluctuates on a large time scale as compared to the other time scales of the complex system under consideration the proposed method extracts the main superstatistical parameters out of a given data set and examines the validity of the superstatistical model assumptions we test the method thoroughly with surrogate data sets then the applicability of the superstatistical approach is illustrated using real experimental data we study two examples velocity time series measured in turbulent taylor couette flows and time series of log returns of the closing prices of some stock market indices

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

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

sets of integers with no large sum free subset answering a question of p erd os from 1965 we show that for every 0 there is a set a of n integers with the following property every set a 0 a with at least 1 n elements contains three distinct elements x y z with x y z contents

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

strict inequalities of critical values in continuum percolation we consider the supercritical finite range random connection model where the points x y of a homogeneous planar poisson process are connected with probability f y x for a given f performing percolation on the resulting graph we show that the critical probabilities for site and bond percolation satisfy the strict inequality p c mathrm site p c mathrm bond we also show that reducing the connection function f strictly increases the critical poisson intensity

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

a superstatistical model of vehicular traffic flow abstract in the analysis of vehicular traffic flow a myriad of techniques have been implemented in this study superstatistics is used in modeling the traffic flow on a highway segment traffic variables such as vehicular speeds volume and headway were collected for three days for the superstatistical approach at least two distinct time scales must exist so that a superposition of nonequilibrium systems assumption could hold when the slow dynamics of the vehicle speeds exhibit a gaussian distribution in between the fluctuations of the system at large one speaks of a relaxation to a local equilibrium these gaussian distributions are found with corresponding standard deviations 1 this translates into a series of fluctuating beta values hence the statistics of statistics superstatistics the traffic flow model has generated an inverse temperature parameter beta distribution as well as the speed distribution this beta distribution has shown that the fluctuations in beta are distributed with respect to a chi square distribution it must be mentioned that two distinct tsallis q values are specified one is time dependent and the other is independent a ramification of these q values is that the highway segment and the traffic flow generate separate characteristics this highway segment in question is not only nonadditive in nature but a nonequilibrium driven system with frequent relaxations to a gaussian

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

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

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

modelling train delays with q exponential functions we demonstrate that the distribution of train delays on the british railway network is accurately described by q exponential functions we explain this by constructing an underlying superstatistical model

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

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 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

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

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

nonuniversality and continuity of the critical covered volume fraction in continuum percolation we establish using mathematically rigorous methods that the critical covered volume fraction cvf for a continuum percolation model with overlapping balls of random sizes is not a universal constant independent of the distribution of the size of the balls in addition we show that the critical cvf is a continuous function of the distribution of the radius random variable in the sense that if a sequence of random variables converges weakly to some random variable then the critical cvf based on these random variables converges to the critical cvf of the limiting random variable

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

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

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

from time series to superstatistics complex nonequilibrium systems are often effectively described by a statistics of a statistics in short a superstatistics we describe how to proceed from a given experimental time series to a superstatistical description we argue that many experimental data fall into three different universality classes chi2 superstatistics tsallis statistics chi2 inverse superstatistics and log normal superstatistics we discuss how to extract the two relevant well separated superstatistical time scales tau and t the probability density of the superstatistical parameter beta and the correlation function for beta from the experimental data we illustrate our approach by applying it to velocity time series measured in turbulent taylor couette flow which is well described by log normal superstatistics and exhibits clear time scale separation

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

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

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

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

towards a dynamical model for wireless sensor networks in this paper we introduce two dynamical models for a broadcast process in wireless sensor networks we obtain a convergent martingale sequence for the two models to our knowledge such martingales were unknown previously we look at the formal models using the formalisms of martingales dynamical systems and markov chains each formalism providing complementary and coherent information with each other the dynamics of both models are comparable and are validated in their domain of application with numerical simulation of wireless sensor networks we make explicit the situations where the models are realistic we also provide a formal analysis of the quasi stationary distribution associated to the markov chain corresponding to the second model proposed

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

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

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

vinogradov s theorem with almost equal summands let 11 20 we prove that every sufficiently large odd integer n can be written as a sum of three primes n p1 p2 p3 with pi n 3 n for i 1 2 3

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

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

determining the complexity of multi component conformal systems a platoon based approach abstract many systems in nature and engineering are composed of subsystems these subsystems may be formed in a linear planar or spatial array a typical example of these formations is a chain of vehicles known as platoon formation in traffic flow platoon formation of vehicles is a linear or planar formation of vehicles where a certain and a constant headway and sideway if applicable is provided in between every and each one of them it is argued in this paper that a well automated platoon formation of vehicles is an extreme case of conformity during this transformation from a many degrees of freedom formation to a solid object tsallis q value is computed to be ranging from one extreme case of q 3 to the other where q 1 when classified in terms of inverse temperatures of clearance fluctuations at one extreme of q 3 one observes unbounded fluctuations in clearance fluctuations so that inverse temperature distributions approach a dirac delta at the origin at the other extreme of q 1 fluctuations in clearance tend to zero asymptotically where a solid structure of agents vehicles emerges the transition from q 3 to q 1 is investigated through synthetic and experimental clearance fluctuations between the cars the results show that during the transition from q 3 to q 1 the platoon loses its many degrees of freedom dof of motion until a solid single object emerges authors assert that the tsallis q value of a platoon of vehicles is limited to 3 q 1

optimization of shape in continuum percolation we consider a version of the boolean or poisson blob continuum percolation model where at each point of a poisson point process in the euclidean plane with intensity a a copy of a given compact convex set a with fixed rotation is placed to each a we associate a critical value c a which is the infimum of intensities a for which the occupied component contains an unbounded connected component it is shown that min c a a convex of area a is attained if a is any triangle of area a and max c a a convex of area a is attained for some centrally symmetric convex set a of area a it turns out that the key result which is also of independent interest is a strong version of the difference body inequality for convex sets in the plane in the plane the difference body inequality states that for any compact convex set a 4 a a a 6 a with equality to the left iff a is centrally symmetric and with equality to the right iff a is a triangle here denotes area and a a is the difference body of a we strengthen this to the following result for any compact convex set a there exist a centrally symmetric convex set c and a triangle t such that c t a and c c a t t with equality to the left iff a is centrally symmetric and to the right iff a is a triangle

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

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

asymmetry in the percolation thresholds of fully penetrable disks with two different radii we perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii as a function of volumetric proportion v the percolation threshold phic v lambda is approximately symmetric about v 0 5 for a fixed ratio lambda of the radii however the difference from symmetry is strongly statistically significant we also improve by an order of magnitude the measurement of the percolation threshold for disks of equal radius phic 0 676 347 5 6

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