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

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

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

convolutions transforms and convex bodies the paper studies convex bodies and star bodies in bbb r n by using radon transforms on grassmann manifolds p cosine transforms on the unit sphere and convolutions on the rotation group of bbb r n it presents dual mixed volume characterizations of i intersection bodies and l p balls which are related to certain volume inequalities for cross sections of convex bodies it considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties convolution techniques are used to derive formulas for mixed volumes mixed surface measures and p cosine transforms they are also used to prove characterizations of geometric functionals such as surface area and dual quermassintegrals

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

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

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

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

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

lp aleksandrov fenchel inequality abstract in this paper we establish l p aleksandrov fenchel inequality as applications we prove a new brunn minkowski type inequality and generalize some interrelated results

higher regularity of global attractors of a weakly dissipative fractional korteweg de vries equation in this paper we study the long time behavior of solutions to a weakly dissipative fractional korteweg de vries kdv equation on the real line r the main difficulty lies in that the dissipative term is a nonlocal operator we overcome this difficulty by the commutator estimates and product estimates associated with fractional laplacian the asymptotical compactness of solution semigroup is proved by the tail estimates finally we conclude the existence of h2 r h5 r global attractor of the weakly dissipative fractional kdv equation

a new type of hopf algebra which is neither commutative nor cocommutative a new determinant cofactor method is used to impose the crucial constraints on the entries of the multiparametric r matrices mentioned by yu i manin 1988 the authors obtain the quotient hopf algebras from the ybzf algebras which are defined by the restricted r matrices a subclass of algebra with the q parameter is also discussed

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

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

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

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

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

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