minimal volume product near hanner polytopes abstract mahler s conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in a fixed dimension it is known that every hanner polytope has the same volume product as the cube or the cross polytope in this paper we prove that every hanner polytope is a strict local minimizer for the volume product in the class of symmetric convex bodies endowed with the banach mazur distance

design of signature sequences for overloaded cdma and bounds on the sum capacity with arbitrary symbol alphabets in this paper we explore some of the fundamentals of synchronous code division multiple access cdma as applied to wireless and optical communication systems under very general settings of any size for the user symbols and the signature matrix entries the channel is modeled by real complex additive noise of arbitrary distribution two problems are addressed the first problem concerns whether uniquely detectable overloaded matrices exist in the absence of additive noise under these general settings and if so whether there are any practical optimum detection algorithms the second one is about the bounds for the sum channel capacity when user data and signature matrices employ any real or complex alphabets finite or infinite in response to the first problem we have developed practical maximum likelihood detection algorithms for overloaded cdma systems for a large class of alphabets in response to the second problem a general theorem has been developed in which the sum capacity lower bounds with respect to the number of users spreading gain and signal to noise ratio can be derived to show the power and utility of the main theorem a number of sum capacity bounds for special cases are evaluated an important conclusion of this paper is that the lower and upper bounds of the sum capacity for small medium size cdma systems depend on both the input and the signature symbols this is contrary to the asymptotic results for large scale systems reported in the literature also confirmed in this paper where the signature symbols and statistics disappear for signature matrices and input vectors with i i d entries furthermore upper and asymptotic bounds are derived and compared to other derivations

smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

a remark on the mahler conjecture local minimality of the unit cube we prove that the unit cube b n is a strict local minimizer for the mahler volume product voln k voln k in the class of origin symmetric convex bodies endowed with the banach mazur distance

geometry of the l_q centroid bodies of an isotropic log concave measure we study some geometric properties of the lq centroid bodies zq of an isotropic log concave measure on r for any 2 6 q 6 n and for e e0 q n 1 we determine the inradius of a random 1 e n dimensional projection of zq up to a constant depending polynomially on e using this fact we obtain estimates for the covering numbers n qb 2 tzq t 1 thus showing that zq is a regular convex body as a consequence we also get an upper bound for m zq

new orlicz affine isoperimetric inequalities abstract the orlicz brunn minkowski theory received considerable attention recently and many results in the l p brunn minkowski theory have been extended to their orlicz counterparts the aim of this paper is to develop orlicz l affine and geominimal surface areas for a single convex body as well as for multiple convex bodies which generalize the l p mixed affine and geominimal surface areas fundamental concepts in the l p brunn minkowski theory our extensions are different from the general affine surface areas by ludwig in 21 moreover our definitions for orlicz l affine and geominimal surface areas reveal that these affine invariants are essentially the infimum supremum of v k l the orlicz mixed volume of k and the polar body of l where l runs over all star bodies and all convex bodies respectively with volume of l equal to the volume of the unit euclidean ball b 2 n properties for the orlicz l affine and geominimal surface areas such as affine invariance and monotonicity are proved related orlicz affine isoperimetric inequalities are also established

a characterization of strongly regular graphs in terms of the largest signless laplacian eigenvalues abstract let g be a simple graph of order n with maximum degree let resp denote the maximum number of common neighbors of a pair of adjacent vertices resp nonadjacent distinct vertices of g let q g denote the largest eigenvalue of the signless laplacian matrix of g we show that q g 4 4 2 1 n 1 2 with equality if and only if g is a strongly regular graph with parameters n

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

the spectral radii of a graph and its line graph the spectral radius of a directed graph is the largest eigenvalue of adjacency matrix of the directed graph we give the relation on the characteristic polynomials of a directed graph and its line graph and obtain sharp bounds on the spectral radius of directed graphs we also give the relation on the spectral radii of a graph and its line graph as a consequence the spectral radius of a connected graph does not exceed that of its line graph except that the graph is a path

sections of convex bodies the generalized busemann petty problem asks if the volume of i dimensional central section of a centrally symmetric convex body in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is smaller than that of another such body is the volume of the body also smaller it is proved that the answer is negative if 2 i n the case of a 2 dimensional section remains open the proof uses techniques in functional analysis and radon transforms on grassmannians it also requires the notion of an i intersection body which generalizes the notion of an intersection body inequalities among the volumes of projection bodies polar projection bodies and their central sections are proved they are related to the maximal slice problem

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

code construction for the t user noiseless adder channel a general method for constructing codes for the t user noiseless adder channel is proposed which achieves the largest sum rate known the authors consider a multiple access communication system in this system t statistically independent sources which use binary block codes c sub 1 c sub t of equal length n simultaneously transmit information via one common channel maintaining synchronization with respect to words and bits the output of the adder channel is the t l ary vector which is the component wise arithmetic sum of the transmitted binary vectors the task of the decoder is to uniquely determine the messages of all users

new lp affine isoperimetric inequalities we prove new lp affine isoperimetric inequalities for all p 8 1 we establish for all p n a duality formula which shows that lp affine surface area of a convex body k equals affine surface area of the polar body k

dual orlicz brunn minkowski theory dual orlicz lϕ affine and geominimal surface areas abstract this paper aims to develop the basic theory for the dual orlicz l affine and geominimal surface areas of star bodies basic properties of these new affine invariants are established moreover related orlicz affine isoperimetric cyclic santalo type and alexandrov fenchel type inequalities are obtained

slicing inequalities for measures of convex bodies abstract we consider the following problem does there exist an absolute constant c so that for every n n every integer 1 k n every origin symmetric convex body l in r n and every measure with non negative even continuous density in r n 1 l c k max h gr n k l h l k n where gr n k is the grassmanian of n k dimensional subspaces of r n and l stands for volume this question is an extension to arbitrary measures in place of volume and to sections of arbitrary codimension k of the slicing problem a major open problem in convex geometry it was proved in 18 19 that 1 holds for arbitrary origin symmetric convex bodies all k and all with c o n in this article we prove inequality 1 with an absolute constant c for unconditional convex bodies and for duals of bodies with bounded volume ratio we also prove that for every 0 1 there exists a constant c c so that inequality 1 holds for every n n every origin symmetric convex body l in r n every measure with continuous density and the codimension of sections k n the proofs are based on a stability result for generalized intersection bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies in the last section we show that for some measures the behavior of minimal sections may be very different from the case of volume

isomorphic properties of intersection bodies we study isomorphic properties of two generalizations of intersection bodies the class ikn of k intersection bodies in rn and the class bpkn of generalized k intersection bodies in rn in particular we show that all convex bodies can be in a certain sense approximated by intersection bodies namely if k is any symmetric convex body in rn and 1 k n 1 then the outer volume ratio distance from k to the class bpkn can be estimated by o v r k bpkn inf c k 1n c bpkn k c cnklogenk where c 0 is an absolute constant next we prove that if k is a symmetric convex body in rn 1 k n 1 and its k intersection body ik k exists and is convex then dbm ik k b2n c k where c k is a constant depending only on k dbm is the banach mazur distance and b2n is the unit euclidean ball in rn this generalizes a well known result of hensley and borell we conclude the paper with volumetric estimates for k intersection bodies

approximate polytope membership queries in the polytope membership problem a convex polytope k in mathbb r d is given and the objective is to preprocess k into a data structure so that given any query point q in mathbb r d it is possible to determine efficiently whether q in k we consider this problem in an approximate setting given an approximation parameter varepsilon the query can be answered either way if the distance from q to k s boundary is at most varepsilon times k s diameter we assume that the dimension d is fixed and k is presented as the intersection of n halfspaces previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by dudley approx theory 10 1974 pp 227 236 and bentley faust and preparata commun acm 25 1982 pp 64 68 the former is optimal in the worst case with respect to space and the latter is optimal with respect to query time we present four main results first we show how

when random proportional subspaces are also random quotients we discuss when a generic subspace of some fixed proportional dimension of a finite dimensional normed space can be isomorphic to a generic quotient of some proportional dimension of another space we show in theorem 4 1 that if this happens for some natural random structures then for any proportion arbitrarily close to 1 the first space has a lot of euclidean subspaces and the second space has a lot of euclidean quotients

upper bounds on the spectral radius of book free and or k2 l free graphs abstract the spectral radius g of a graph g is the largest eigenvalue of its adjacency matrix let b k denote a book with k pages in this paper we generalize a result of lu et al m lu h liu f tian a new upper bound for the spectral radius of graphs with girth at least 5 linear algebra appl 414 2006 512 516 on the upper bound for the spectral radius of connected graphs with girth at least 5 to connected b k 1 k 2 l 1 free graphs g of order n with maximum degree as follows g k l k l 2 4 4 l n 1 2 with equality if and only if g is a strongly regular graph with parameters k l this implies sharp upper bounds for book free or k 2 l free graphs

lattice free sets multi branch split disjunctions and mixed integer programming in this paper we study the relationship between valid inequalities for mixed integer sets lattice free sets associated with these inequalities and the multi branch split cuts introduced by li and richard discret optim 5 724 734 2008 by analyzing n dimensional lattice free sets we prove that for every integer n there exists a positive integer t such that every facet defining inequality of the convex hull of a mixed integer polyhedral set with n integer variables is a t branch split cut we use this result to give a finite cutting plane algorithm to solve mixed integer programs we also show that the minimum value t for which all facets of polyhedral mixed integer sets with n integer variables can be generated as t branch split cuts grows exponentially with n in particular when n 3 we observe that not all facet defining inequalities are 6 branch split cuts

bounds on the laplacian spectral radius of graphs the spectral radius of a graph is the largest eigenvalue of adjacency matrix of the graph and its laplacian spectral radius is the largest eigenvalue of the laplacian matrix which is the difference of the diagonal matrix of vertex degrees and the adjacency matrix some sharp bounds are obtained for the laplacian spectral radii of connected graphs as consequences some sharp upper bounds of the nordhaus gaddum type are also obtained for the sum of laplacian spectral radii of a connected graph and its connected complement

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

the volume product of convex bodies with many hyperplane symmetries mahler s conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume we confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense their hyperplanes of symmetries have a one point intersection moreover we obtain improved sharp lower bounds for classes of convex bodies which are invariant by certain reflection groups namely direct products of the isometry groups of regular polytopes

the spectral turan problem about graphs with no 6 cycle abstract a spectral version of extremal graph theory problem is as follows what is the maximum spectral radius of an h free graph of order n nikiforov posed a conjecture on spectral extremal value of cycles in the past decade much attention has been paid to this conjecture and other questions on spectral extremal graphs in order to approach to nikiforov s conjecture we provide a new conjecture on spectral extremal value involving fans and wheels moreover we show it is true for k 2

affine quermassintegrals of random polytopes abstract a question related to some conjectures of lutwak about the affine quermassintegrals of a convex body k in r n asks whether for every convex body k in r n and all 1 k n k k vol n k 1 n g n k vol k p f k n d n k f 1 k n c n k where c 0 is an absolute constant we provide an affirmative answer for some broad classes of random polytopes we also discuss upper bounds for k k when k b 1 n the unit ball of l 1 n and explain how this special instance has implications for the case of a general unconditional convex body k

lowm estimates on coordinate subspaces letkbe a symmetric convex body inrn it is well known that for every 0 1 there exists a subspacefofrnwith dim f 1 n such thatpf k c mkdn f where pfdenotes the orthogonal projection ontof consider a fixed coordinate system inrn we study the question whether an analogue of can be obtained when one is restricted to choosefamong the coordinate subspacesr 1 n with 1 n we prove several coordinate versions of in terms of the cotype 2 constant of the volume ratio and other parameters ofk the basic source of our estimates is an exact coordinate analogue of in the ellipsoidal case applications to the computation of the number of lattice points inside a convex body are considered throughout the paper

further results on coding for t user multiple access channels corresp coding schemes for the noiseless t user binary adder multiple access channel are discussed a class of uniquely decodable codes of arbitrary length satisfying a specific recursive relationship and which are asymptotically good is explicitly constructed

symmetric mahler s conjecture for the volume product in the 3 dimensional case we prove mahler s conjecture concerning the volume product of centrally symmetric convex bodies in rn in the case where n 3 more precisely we show that for every 3 dimensional centrally symmetric convex body k r3 the volume product k k is greater than or equal to 32 3 with equality if and only if k or k is a parallelepiped

a family of k 1 ary signature codes for noisy multiple access adder channel a coding scheme of k 1 ary error correcting signature codes for a noisy multiple access adder channel is proposed given a signature matrix a and a difference matrix d d d a priori a larger signature matrix is obtained by replacing each element in the hadamard matrix with a or d or d depending on the values of the elements and their locations in the hadamard matrix the set of rows in the proposed matrix gives an error correcting signature code introducing a difference matrix makes it possible to construct an error correcting signature code whose sum rate is increased with an increase in the order of the hadamard matrix either binary or non binary signature codes are constructed when the pairs of matrices a and d are given

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

stability of the blaschke santalo and the affine isoperimetric inequality abstract a stability version of the blaschke santalo inequality and the affine isoperimetric inequality for convex bodies of dimension n 3 is proved the first step is the reduction to the case when the convex body is o symmetric and has axial rotational symmetry this step works for related inequalities compatible with steiner symmetrization secondly for these convex bodies a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established

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

how often is a random quantum state k entangled the set of trace preserving positive maps acting on density matrices of size d forms a convex body we investigate its nested subsets consisting of kpositive maps where k 2 d working with the measure induced by the hilbert schmidt distance we derive asymptotically tight bounds for the volumes of these sets our results strongly suggest that the inner set of k 1 positive maps forms a small fraction of the outer set of k positive maps these results are related to analogous bounds for the relative volume of the sets of k entangled states describing a bipartite d d system

some functional inverse santalo inequalities abstract let f r n r be a log concave function and for z r n define f z y inf x r n e x z y z f x for every y r n we discuss the problem of finding a sharp lower bound to the product p f inf z r n r n f x d x r n f z y d y we prove that if n 1 then p f e and characterize the case of equality the same method allows to give a new simple proof of the fact that if f is sign invariant then for all n p f 4 n these inequalities are functional versions with exact lower bounds of the so called inverse santalo inequality for convex bodies that we state and discuss as conjectures

generalized t user codes for multiple access channels corresp large classes of t user codes for the binary adder channel are shown to be easily constructed by generalizing some previous results in two basic ways the first is an iterative procedure the second is by introducing an equivalence relation on such codes decoding algorithms are presented in both cases the number of codes in the equivalence class of some previously defined codes is determined exactly

walks and the spectral radius of graphs abstract given a graph g write g for the largest eigenvalue of its adjacency matrix g for its clique number and wk g for the number of its k walks we prove that the inequalities w q r g w q g r g g 1 g w r g hold for all r 0 and odd q 0 we also generalize a number of other bounds on g and characterize semiregular and pseudo regular graphs in spectral terms

a bound on the laplacian spread which is tight for strongly regular graphs abstract the laplacian spread of a graph is the difference between the largest eigenvalue and the second smallest eigenvalue of the laplacian matrix of the graph for laplacian matrices of graphs we find their upper bounds of largest eigenvalues lower bounds of second smallest eigenvalues and upper bounds of laplacian spreads the strongly regular graphs attain all the above bounds some other extremal graphs are also provided

the flatness theorem for nonsymmetric convex bodies via the local theory of banach spaces let l be a lattice in rn and k a convex body disjoint from l the classical flatness theorem asserts that then w k l the l width of k does not exceed some bound depending only on the dimension n this fact was later found relevant to questions in integer programming kannan and lovasz 1988 showed that under the above assumptions w k l cn2 where c is a universal constant banaszczyk 1996 proved that w k l cn 1 log n if k has a centre of symmetry in the present paper we show that w k l cn3 2 for an arbitrary k it is conjectured that the exponent 3 2 may be replaced by 1 perhaps at the cost of a logarithmic factor we prove that for some naturally arising classes of bodies

a new upper bound for the spectral radius of graphs with girth at least 5 let g be a simple connected graph with v g v1 v 2 v n and girth at least 5 let be the maximum degree of g in this paper we present a new sharp upper bound for the spectral radius of g as follows

neighborhoods on the grassmannian of marginals with bounded isotropic constant abstract we show that for any isotropic log concave probability measure on r n for every e 0 every 1 k n and any e g n k there exists f g n k with d e f e and l f c e

recursive constructions of detecting matrices for multiuser coding a unifying approach detecting matrices are a class of combinatorial objects originated from the coin weighing problem of soderberg and shapiro in the early 1960s in this paper various known recursive construction techniques for binary bipolar and ternary detecting matrices are reexamined in a unifying framework new general recursive constructions of detecting matrices which include previous recursive constructions as special cases are derived such matrices find applications in multiuser coding since they are equivalent to a certain class of uniquely decodable multiuser codes for the binary adder channel interestingly it is found that among the three kinds of detecting matrices ternary detecting matrices are of fundamental significance from the combinatorial theoretic as well as from the multiuser coding application point of view

volume ratio and other s numbers of operators related to local properties of banach spaces abstract volume ratio numbers are studied and inequalities relating them to other s numbers of operators are established characterizations of spaces of weak cotype 2 in terms of s numbers and of operator ideal norms are obtained

sharp upper bounds on the spectral radius of graphs abstract let g be a simple connected graph with n vertices m edges and degree sequence d 1 d 2 d n the spectral radius g of graph g is the largest eigenvalue of its adjacency matrix in this paper we present some sharp upper bounds of the spectral radius in terms of the degree sequence of graphs

from symplectic measurements to the mahler conjecture in this note we link symplectic and convex geometry by relating two seemingly different open conjectures a symplectic isoperimetric type inequality for convex domains and mahler s conjecture on the volume product of centrally symmetric convex bodies more precisely we show that if for convex bodies of fixed volume in the classical phase space the hofer zehnder capacity is maximized by the euclidean ball then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies

bounds on graph eigenvalues i abstract we prove three results about the spectral radius g of a graph g a let t r n be the r partite turan graph of order n if g is a k r 1 free graph of order n then g t r n unless g t r n b for most irregular graphs g of order n and size m g 2 m n 1 2 m 2 n c let 0 k l if g is a graph of order n with no k 2 k k 1 and no k 2 l 1 then g min g k l 1 k l 1 2 4 l n 1 2

flag numbers and floating bodies abstract we investigate weighted floating bodies of polytopes we show that the weighted volume depends on the complete flags of the polytope this connection is obtained by introducing flag simplices which translate between the metric and combinatorial structure our results are applied in spherical and hyperbolic space this leads to new asymptotic results for polytopes in these spaces we also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in euclidean space