estimates for measures of lower dimensional sections of convex bodies abstract we present an alternative approach to some results of koldobsky on measures of sections of symmetric convex bodies which allows us to extend them to the not necessarily symmetric setting we prove that if k is a convex body in r n with 0 int k and is a measure on r n with a locally integrable non negative density g on r n then k c n k k max f g n n k k f k k n for every 1 k n 1 also if is even and log concave and if k is a symmetric convex body in r n and d is a compact subset of r n such that k f d f for all f g n n k then k c k l n k k d where l s is the maximal isotropic constant of a convex body in r s our method employs a generalized blaschke petkantschin formula and estimates for the dual affine quermassintegrals

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

inequalities for hadwiger s harmonic quermassintegrals let j ff denote the set of convex bodies compact sets with nonempty interiors in euclidean n space ir and let denote the subset of t tm that contains the centered centrally symmetric with respect to the origin bodies in let g n i denote the grassmann manifold of dimensional subspaces of ir and let 1i denote the usual haar measure on g n i normalized so that i g n i 1 for k ff and g n i we use ki to denote the image of the orthogonal projection of k onto and v k to denote the dimensional volume of the body ki in the quermassintegrals wo k wi k w k of a convex body k in r can be defined by letting wo k v k the volume of k w k co the volume of the unit ball b in ir and for i such that 0 i n by

cohomological degrees and hilbert functions of graded modules making use of the recent construction of cohomological degrees functions we give several estimates on the relationship between number of generators and degrees of ideals and modules with applications to hilbert functions they extend results heretofore known from generalized cohen macaulay local rings to nearly arbitrary local rings the rules of computation these functions satisfy enables comparison with castelnuovo mumford s regularity in the graded case as application we derive sharp improvements on predicting the outcome of effecting noether normalizations in tangent cones

busemann s intersection inequality in hyperbolic and spherical spaces abstract busemann s intersection inequality asserts that the only maximizers of the integral s n 1 k n d among all convex bodies of a fixed volume in r n are centered ellipsoids we study this question in the hyperbolic and spherical spaces as well as general measure spaces

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

disk envelopes of functions ii given a suslin function on a domainyin a banach spacexone can define two envelopes of related to the pluripotential theory the first one denoted byp is the supremum of all plurisubharmonic functions less than the value of the other one denoted byd at some pointzis equal to the infimum of integrals over the boundaries of all closed analytic disks centered atz in this paper we study the relationship between these envelopes we show thatd p for a large class of functions on domains in banach spaces wherep is the upper regularization ofp ifx cn then we prove thatd is always equal top

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

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

invariant valuations on star shaped sets the brunn minkowski theory of convex bodies and mixed volumes has provided many tools for solving problems involving projections and valuations of compact convex sets in euclidean space among the most beautiful results of twentieth century convexity is hadwiger s characterization theorem for the elementary mixed volumes quermassintegrals see 3 5 9 hadwiger s characterization leads to effortless proofs of numerous results in geometric convexity including mean projection formulas for convex bodies 13 p 294 and various kinematic formulas 7 12 14 15 hadwiger s theorem also provides a connection between rigid motion invariant set functions and symmetric polynomials 1 7 recently advancements have been made in a theory introduced by lutwak 8 that is dual to the brunn minkowski theory a theory tailored for dealing with analogous questions involving star shaped sets and intersections with subspaces see also 2 4 6 in the dual theory convex bodies are replaced by star shaped sets and support functions are replaced by radial functions hadwiger s characterizaton theorem is of such fundamental importance that any candidate for a dual theory must possess a dual analogue however the dual theory in its original form was not sufficiently rich to be able to accommodate a dual of hadwiger s theorem in 6 it was shown that the natural setting for the dual theory is larger than that envisioned by previous investigators by defining the dual topology on star shaped sets in terms of the l topology on the space of n integrable functions on the unit sphere the author was able to extend the dual theory to a broad class of star shaped sets called l stars many new theorems can be proved within this larger framework including a hadwiger style classification theorem for continuous valuations on star shaped sets that are homogeneous with respect to dilation in the present paper we discard the stringent requirement of homogeneity and continue with classification theorems for continuous valuations on article no ai971601

convexity argument for monotone potential operators and its application modelling of many applied problems leads to the nonlinear equations elliptic type with monotone potential operators 6 11 some methods have been developed for di erent classes of nonlinear operators to study such qualitative questions as existence uniqueness linearization of the nonlinear operator and convergence of an approximate solution 2 7 11 in some cases it is possible to construct an abstract iteration technique for linearization of the nonlinear operator and to prove the convergence of an approximate solution in this paper we prove an abstract iterative scheme for monotone potential operators by using the convexity argument note that in the literature di erent convexity arguments or inequalities have been applied to problems of mathematical physics 14 in particular in 9 10 the logarithmic convexity argument was used to establish continuous dependence on the initial data for the cauchy problem the convexity argument introduced here is used as a su cient condition for the convergence of the abstract iteration scheme for monotone potential operators then using the result the convergence of the iteration scheme is proved for concrete nonlinear problems the paper is organized as follows in section 2 we introduce the functional inequality which is called the convexity argument then we consider an abstract operator equation and by using the argument we prove the strong convergence of the iteration

tomography of convex and star bodies new definitions of a star body and its radial function are given these are used in studying sections of convex and star bodies in conjunction with the dual mixed volumes of lutwak the spherical radon transform and the i chord functions when the body contains the origin in its interior and i 0 the latter equals the sum of the values of the ith power of the radial function at antipodal points in the unit sphere special cases of results obtained include the following a simple construction is given of non spherical convex bodies of constant i section i e such that each i dimensional section through an interior point has the same i dimensional volume it is shown that two star bodies containing the origin in their interiors and with analytic boundaries must be equal up to reflection in the origin if the volumes of their sections by i dimensional subspaces for two different values of i are equal examples are given to show that the last statement is no longer necessarily true if the boundaries are only infinitely differentiable or if the bodies are convex polytopes it is also proved that a star body which has constant i section for two different values of i with respect to the same interior point must be a ball centered at that point this answers the dual of the old question as to whether there exist non spherical convex bodies of constant width and constant brightness new results are also obtained concerning the determination of convex polytopes by the volumes of their sections through one or more points

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

star valuations and dual mixed volumes since its creation by brunn and minkowski what has become known as the brunn minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data the machinery of the brunn minkowski theory includes mixed volumes of minkowski symmetrization techniques such as those of steiner and blaschke isoperimetric inequalities such as the brunn minkowski minkowski and aleksandrov fenchel inequalities integral transforms such as the cosine transform and important auxiliary bodies associated with these transforms such as minkowski s projection bodies schneider s recent book 22 on the brunn minkowski theory is the best available introduction to the subject while the brunn minkowski theory has proven to be of enormous value in answering inverse questions regarding projections of convex bodies onto subspaces the theory has been of little value in answering inverse questions with data regarding intersections with subspaces however recent advancements have been made in the development of a dual brunn minkowski theory 3 4 5 6 8 9 11 13 16 23 26 27 28 29 which has been tailored specifically for dealing with such questions in contrast to the brunn minkowski theory in the dual theory convex bodies are replaced by star shaped sets and projections onto subspaces are replaced by intersections with subspaces the machinery of the dual theory includes dual mixed volumes introduced by lutwak 15 16 dual isoperimetric inequalities 16 and important auxiliary bodies known as intersection bodies first introduced by busemann for special centered convex bodies in 1 later defined for star shaped sets by lutwak 16 see also 9 a comprehensive introduction to geometric tomography including the dual brunn minkowski theory may be found in gardner s book 17 unfortunately there still remain fundamental and foundational problems with the dual brunn minkowski theory one of the most beautiful and article no 0048

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

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

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 existence uniqueness and convergence of approximate solution of boundary value problems related to the nonlinear operator au k u 2 u g u we study the problems of solvability and linearization for the nonlinear boundary value problems with nonlinear operator au k u 2 u g u solvability in h 1 a b of the problems is obtained by using monotone potential operator theory and browder minty theorem sufficient conditions for the solvability are obtained in explicit form for the linearization of the considered nonlinear problems monotone iterative scheme is developed the scheme permits use of the variational finite difference scheme for the numerical solution the considered nonlinear problems sufficient conditions for the convergence of the iteration method are presented computational experiments illustrate high accuracy of the presented method

subsets of full measure in a generic submanifold in mathbb c n are non plurithin in this paper we prove that if i subset m is a subset of measure 0 in a c 2 smooth generic submanifold m subset mathbb c n then m setminus i is non plurithin at each point of m in mathbb c n this result improves a previous result of a edigarian and j wiegerinck who considered the case where i is pluripolar set contained in a c 1 smooth generic submanifold m subset mathbb c n edigarian and wiegernick in math z 266 2 393 398 2010 the proof of our result is essentially different

lutwak petty projection inequalities for minkowski valuations and their duals abstract lutwak s volume inequalities for polar projection bodies of all orders are generalized to polarizations of minkowski valuations generated by even zonal measures on the euclidean unit sphere this is based on analogues of mixed projection bodies for such minkowski valuations and a generalization of the notion of centroid bodies a new integral representation is used to single out lutwak s inequalities as the strongest among these families of inequalities which in turn are related to a conjecture on affine quermassintegrals in the dual setting a generalization of volume inequalities for intersection bodies of all orders by leng and lu is proved these results are related to grinberg s inequalities for dual affine quermassintegrals

divergence for s concave and log concave functions abstract we prove new entropy inequalities for log concave and s concave functions that strengthen and generalize recently established reverse log sobolev and poincare inequalities for such functions this leads naturally to the concept of f divergence and in particular relative entropy for s concave and log concave functions we establish their basic properties among them the affine invariant valuation property applications are given in the theory of convex bodies

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

norm estimates for k plane transforms and geometric inequalities abstract the article is devoted to remarkable interrelation between the norm estimates for k plane transforms in weighted and unweighted l p spaces and geometric integral inequalities for cross sections of measurable sets in r n a similar interrelation is studied for more general j plane to k plane transforms on affine grassmannians and their compact modifications the article contains a series of new integral geometric inequalities with sharp constants explicit equalities conjectures and open problems

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

the dual brunn minkowski theory for bounded borel sets dual affine quermassintegrals and inequalities abstract this paper develops a significant extension of e lutwak s dual brunn minkowski theory originally applicable only to star shaped sets to the class of bounded borel sets the focus is on expressions and inequalities involving chord power integrals random simplex integrals and dual affine quermassintegrals new inequalities obtained include those of isoperimetric and brunn minkowski type a new generalization of the well known busemann intersection inequality is also proved particular attention is given to precise equality conditions which require results stating that a bounded borel set almost all of whose sections of a fixed dimension are essentially convex is itself essentially convex

an inverse problem for an elastoplastic medium in this paper within the range of the so called deformation theory of plasticity the determination of unknown coefficients in a nonlinear system of equilibrium equations from over specified data measured at the boundary is considered this inverse problem is reformulated as a minimization problem for a certain functional it is shown that this functional has at least one solution quasisolution in an admissible class of coefficients then the existence of an exact solution of inverse problem is proved a numerical algorithm and examples related to direct and inverse problems are presented

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

new sine ellipsoids and related volume inequalities abstract corresponding to the legendre ellipsoid and the lyz ellipsoid two new sine ellipsoids are introduced in this paper these four ellipsoids are closely related in the pythagorean relation and duality several volume inequalities and the valuation properties are obtained for two new ellipsoids