a characterization of dual quermassintegrals and the roots of dual steiner polynomials abstract let m 1 r 0 0 r 1 r m be a tuple of distinct real numbers and n 2 we provide a characterization of those tuples 0 1 m of real numbers such that there exist n dimensional star bodies k l with w r j k l j j 0 m where w r k l denotes the r th dual relative quermassintegral of k and l this may be regarded as an analogue within the dual brunn minkowski theory of shephard s classification of quermassintegrals of two convex bodies it turns out that the characterization of dual quermassintegrals is related to the moment problem and based on this relation we also derive new determinantal inequalities among the dual quermassintegrals moreover this characterization will be the key tool in order to investigate structural properties of the set of roots of dual steiner polynomials of star bodies

on the local equatorial characterization of zonoids and intersection bodies abstract in this paper we show that there is no local equatorial characterization of zonoids in odd dimensions this gives a negative answer to the conjecture posed by w weil in 1977 and shows that the local equatorial characterization of zonoids may be given only in even dimensions in addition we prove a similar result for intersection bodies and show that there is no local characterization of these bodies

characterization of dual mixed volumes via polymeasures we prove a characterization of the dual mixed volume in terms of functional properties of the polynomial associated with it to do this we use tools from the theory of multilinear operators on spaces of continuous functions along the way we reprove with these same techniques a recently found characterization of the dual mixed volume

derivatives of the lp cosine transform abstract the l p cosine transform of an even continuous function f c e s n 1 is defined by h x s n 1 x p f d x r n it is shown that if p is not an even integer then all partial derivatives of even order of h x up to order p 1 including p 1 if p is an odd integer exist and are continuous everywhere in r n 0 as a result of the corresponding differentiation formula we show that if f is a positive bounded function and p 1 then h 1 p is a support function of a convex body whose boundary has everywhere positive gauss kronecker curvature

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

the dual minkowski problem for symmetric convex bodies abstract the dual minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q th dual curvature measure of an origin symmetric convex body in r n a full solution to this is given when 1 q n the necessary and sufficient conditions turn out to be an explicit measure concentration condition to obtain the results a variational approach is used where the functional is the sum of a dual quermassintegral and an entropy integral the proof requires two crucial estimates the first is an estimate of the entropy integral which is obtained by using a spherical partition the second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body

lp dual curvature measures abstract a new family of geometric borel measures on the unit sphere is introduced special cases include the l p surface area measures which extend the classical surface area measure of aleksandrov and fenchel jessen and l p integral curvature which extends alkesandrov s integral curvature in the l p brunn minkowski theory it also includes the dual curvature measures which are the duals of federer s curvature measures in the dual brunn minkowski theory this partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes

intersection bodies and generalized cosine transforms the paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis a new concept of intersection body that unifies some known classes of geometric objects is introduced a parallel between trace theorems in function theory restriction onto lower dimensional subspaces of the spherical radon transforms and the generalized cosine transforms and sections of intersection bodies is established new integral formulas for different classes of cosine transforms are obtained and examples of intersection bodies are given we also revisit some known facts in this area and give them new simple proofs

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

comparison of volumes of convex bodies in real complex and quaternionic spaces abstract the classical busemann petty problem 1956 asks whether origin symmetric convex bodies in r n with smaller hyperplane central sections necessarily have smaller volumes it is known that the answer is affirmative if n 4 and negative if n 4 the same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning we give unified analysis of this circle of problems in real complex and quaternionic n dimensional spaces all cases are treated simultaneously in particular we show that the busemann petty problem in the quaternionic n dimensional space has an affirmative answer if and only if n 2 the method relies on the properties of cosine transforms on the unit sphere we discuss possible generalizations

using g algebras for schur index computation abstract conditions for the existence of a bijective character correspondence that preserves schur indices are given based on the theory of central simple g algebras over fields of characteristic 0 it is then shown how this applies to schur index computation in the case of finite solvable groups

if you can hide behind it can you hide inside it let k and l be compact convex sets in rn suppose that for a given dimension 1 d n 1 every d dimensional orthogonal projection of l contains a translate of the corresponding projection of k does it follow that the original set l contains a translate of k in other words if k can be translated to hide behind l from any perspective does it follow that k can hide inside l a compact convex set l is defined to be d decomposable if l is a direct minkowski sum affine cartesian product of two or more convex bodies each of dimension at most d a compact convex set l is called d reliable if whenever each d dimensional orthogonal projection of l contains a translate of the corresponding d dimensional projection of k it must follow that l contains a translate of k it is shown that for 1 d n 1 1 d decomposability implies d reliability 2 a compact convex set l in rn is d reliable if and only if for all m d 2 no m unit normals to regular boundary points of l form the outer unit normals of an m 1 dimensional simplex 3 smooth convex bodies are not d reliable 4 a compact convex set l in rn is 1 reliable if and only if l is 1 decomposable i e a parallelotope 5 a centrally symmetric compact convex set l in rn is 2 reliable if and only if l is 2 decomposable however there are non centered 2 reliable convex bodies that are not 2 decomposable as a result of 5 above the only reliable centrally symmetric covers in r3 from the perspective of 2 dimensional shadows are the affine convex cylinders prisms however in dimensions greater than 3 it is shown that 3decomposability is only sufficient and not necessary for l to cover reliably with respect to 3 shadows even when l is assumed to be centrally symmetric consider two compact convex subsetsk and l of n dimensional euclidean space suppose that for a given dimension 1 d 0 is sufficiently small but no longer fits inside d the same construction works for any set k inscribed in d and having strictly less than unit diameter another received by the editors june 4 2009 2000 mathematics subject classification primary 52a20 c 2011 daniel a klain

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

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

volume inequalities of convex bodies from cosine transforms on grassmann manifolds abstract the l p cosine transform on grassmann manifolds naturally induces finite dimensional banach norms whose unit balls are origin symmetric convex bodies in r n reverse isoperimetric type volume inequalities for these bodies are established which extend results from the sphere to grassmann manifolds

an extension of a result by lonke to intersection bodies in this paper we prove that intersection bodies cannot be direct sums using fourier analytic techniques this extends a result by lonke we also prove a necessary regularity condition and a convexity condition for a body of revolution to be an intersection body of a star body

radial continuous rotation invariant valuations on star bodies we characterize the positive radial continuous and rotation invariant valuations v defined on the star bodies of rn as the applications on star bodies which admit an integral representation with respect to the lebesgue measure that is v k sn 1 k dm where is a positive continuous function k is the radial function associated to k and m is the lebesgue measure on sn 1 as a corollary we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals

on sections of convex bodies in hyperbolic space abstract we consider two problems on sections of convex bodies in hyperbolic space the first one is a modified version of the busemann petty problem we look at conditions that guarantee a positive answer to this problem in all dimensions the second problem is an analogue of a result of makai martini and odor about origin symmetry if in every direction the parallel section function has a critical value at zero then the body is origin symmetric for both problems we use fourier transform techniques

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

the generalized ricci flow for three dimensional manifolds with one killing vector we consider three dimensional 3d flow equations inspired by the renormalization group rg equations of string theory with a three dimensional target space by modifying the flow equations to include a u 1 gauge field and adding carefully chosen de turck terms we are able to extend recent two dimensional results of bakas to the case of a 3d riemannian metric with one killing vector in particular we show that the rg flow with de turck terms can be reduced to two equations the continual toda flow solved by bakas plus its linearizaton we find exact solutions which flow to homogeneous but not always isotropic geometries

detecting symmetry in star bodies abstract we use fourier transform techniques to prove a result on detecting symmetry in convex and star bodies with the help of conical sections our methods also allow us to give a new proof of the well known theorem of makai martini and odor about maximal hyperplane sections passing through the same point

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

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

local and equatorial characterization of unit balls of subspaces of lp p 0 and properties of the generalized cosine transform abstract in this paper we show that there is no local equatorial characterization of bodies that embed in l p in odd dimensions for all p not even 0 p however bodies that embed in l p for p odd are local equatorially characterizable provided that the dimension is even but not locally characterizable in general this extends results given in panina 1988 13 goodey and weil 1992 5 and nazarov ryabogin and zvavitch 2008 12 concerning the local equatorial characterization of zonoids and intersection bodies these results were for bodies that embed in l 1 and l 1 respectively

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

covering shadows with a smaller volume for n 2 a construction is given for convex bodies k and l in rn such that the orthogonal projection lu onto the subspace u contains a translate of ku for every direction u while the volumes of k and l satisfy vn k vn l a more general construction is then given for n dimensional convex bodies k and l such that each orthogonal projection l onto a k dimensional subspace contains a translate of k while the mth intrinsic volumes of k and l satisfy vm k vm l for all m k for each k 1 n we then define the collection cn k to be the closure under the hausdorff topology of all blaschke combinations of suitably defined cylinder sets prisms it is subsequently shown that if l cn k and if the orthogonal projection l contains a translate of k for every k dimensional subspace of rn then vn k vn l the families cn k called k cylinder bodies of rn form a strictly increasing chain cn 1 cn 2 cn n 1 cn n where cn 1 is precisely the collection of centrally symmetric compact convex sets in rn while cn n is the collection of all compact convex sets in rn members of each family cn k are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes and members of cn k are shown to satisfy certain geometric inequalities related open questions are also posed

capacities surface area and radial sums a dual capacitary brunn minkowski inequality is established for the n 1 capacity of radial sums of star bodies in rn this inequality is a counterpart to the capacitary brunn minkowski inequality for the p capacity of minkowski sums of convex bodies in rn 1 p n proved by borell colesanti and salani when n 3 the dual capacitary brunn minkowski inequality follows from an inequality of bandle and marcus but here a new proof is given that provides an equality condition note that when n 3 the n 1 capacity is the classical electrostatic capacity a proof is also given of both the inequality and a different equality condition when n 2 the latter case requires completely different techniques and an understanding of the behavior of surface area perimeter under the operation of radial sum these results can be viewed as showing that in a sense n 1 capacity has the same status as volume in that it plays the role of its own dual set function in the brunn minkowski and dual brunn minkowski theories

the dual orlicz brunn minkowski theory this paper introduces the dual orlicz brunn minkowski theory for star sets a radial orlicz addition of two or more star sets is proposed and a corresponding dual orlicz brunn minkowski inequality is established based on a radial orlicz linear combination of two star sets a formula for the dual orlicz mixed volume is derived and a corresponding dual orlicz minkowski inequality proved the inequalities proved yield as special cases the precise duals of the conjectured log brunn minkowski and log minkowski inequalities of boroczky lutwak yang and zhang a new addition of star sets called radial m addition is also introduced and shown to relate to the radial orlicz addition

on the lp dual minkowski problem abstract the l p dual curvature measure was introduced by lutwak yang zhang in an attempt to unify the l p brunn minkowski theory and the dual brunn minkowski theory the characterization problem for l p dual curvature measure called the l p dual minkowski problem is a fundamental problem in this unifying theory the l p dual minkowski problem contains the l p minkowski problem and the dual minkowski problem two major problems in modern convex geometry that remain open in general in this paper existence results on the l p dual minkowski problem in the weak sense will be provided moreover existence and uniqueness of the solution in the smooth category will also be demonstrated

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

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

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

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

on polynomially integrable convex bodies abstract an infinitely smooth convex body in r n is called polynomially integrable of degree n if its parallel section functions are polynomials of degree n we prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids if n n 1 this is in contrast with the case of even dimensions and the case of odd dimensions with n n 1 where such bodies do not exist as it was recently shown by agranovsky

generalizations of the busemann petty problem for sections of convex bodies abstract we present generalizations of the busemann petty problem for dual volumes of intermediate central sections of symmetric convex bodies it is proved that the answer is negative when the dimension of the sections is greater than or equal to 4 for two three dimensional sections both negative and positive answers are given depending on the orders of dual volumes involved and certain cases remain open for bodies of revolution a complete solution is obtained in all dimensions

radial continuous valuations on star bodies we show that a radial continuous valuation defined on the n dimensional star bodies extends uniquely to a continuous valuation on the n dimensional bounded star sets moreover we provide an integral representation of every such valuation in terms of the radial function which is valid on the dense subset of the simple borel star sets along the way we also show that every radial continuous valuation defined on the n dimensional star bodies can be decomposed as a sum v v v where both v and v are positive radial continuous valuations

a problem of klee on inner section functions of convex bodies in 1969 vic klee asked whether a convex body is uniquely determined up to translation and reflection in the origin by its inner section function the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction we answer this question in the negative by constructing two infinitely smooth convex bodies of revolution about the xn axis in r n n 3 one origin symmetric and the other not centrally symmetric with the same inner section function moreover the pair of bodies can be arbitrarily close to the unit ball

poisson polyhedra in high dimensions abstract the zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated as the space dimension tends to infinity the model includes the zero cell of stationary and isotropic poisson hyperplane tessellations as well as the typical cell of a stationary poisson voronoi tessellation as special cases it is shown that asymptotically in the space dimension with overwhelming probability these cells satisfy the hyperplane conjecture if the distance exponent and the intensity parameter are suitably chosen dimension dependent functions also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed in the background are new identities linking the f vector of the zero cell to certain dual intrinsic volumes

local properties of intertwining operators on the sphere blaschke s original question regarding the local determination of zonoids or projection bodies has been the subject of much research over the years in recent times this research has been extended to include intersection bodies and it has been shown that neither zonoids nor intersection bodies have local characterizations however it has also been proved that both these classes of bodies admit equatorial characterizations in odd dimensions but not in even dimensions the proofs of these results were mostly analytic using properties of associated spherical integral transforms the cosine transform and the radon transform here we elaborate a general principle showing that such local or equatorial characterization problems are equivalent to corresponding support properties of the spherical operators we discuss this within a general framework for intertwining operators on c functions and apply the results to further geometric constructions namely to certain mean section bodies to lq centroid bodies and to k intersection bodies