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

universal features in the growth dynamics of religious activities we quantify and analyze the growth dynamics of a religious group in 140 countries for a 47 year period 1959 2005 we find that i the distribution of annual logarithmic growth rates exhibits the same functional form for distinct size scales and ii the standard deviation of growth rates scales with size as a power law both findings hold for distinct measures of religious activity these results are in surprising agreement with those found in the study of economic activities and scientific research suggesting that religious activities are governed by universal growth mechanisms we also compare the empirical findings on religious activities with the predictions of general models recently proposed in the context of complex organizations our findings should provide useful information for a better understanding of the mechanisms governing the growth of religion

generalized intersection bodies abstract we study the structures of two types of generalizations of intersection bodies and the problem of whether they are in fact equivalent intersection bodies were introduced by lutwak and played a key role in the solution of the busemann petty problem a natural geometric generalization of this problem considered by zhang led him to introduce one type of generalized intersection bodies a second type was introduced by koldobsky who studied a different analytic generalization of this problem koldobsky also studied the connection between these two types of bodies and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized busemann petty problem we show that these classes share many identical structural properties proving the same results using integral geometry techniques for zhang s class and fourier transform techniques for koldobsky s class using a functional analytic approach we give several surprising equivalent formulations for the equivalence problem which reveal a deep connection to several fundamental problems in the integral geometry of the grassmann manifold

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

optimal mass transport for higher dimensional adaptive grid generation in this work we describe an approach for higher dimensional adaptive grid generation based on solving the l 2 monge kantorovich problem mkp which is a special case of the classical optimal mass transportation problem two methods are developed for computing the coordinate transformation used to define the grid adaptation for the first method the transformation is determined by solving a parabolic monge ampere equation for a steady state solution for the second method the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the l 2 mkp several numerical experiments are presented to demonstrate the performance of the mkp methods and to compare them with some related adaptive grid methods the experimental results demonstrate that the mkp methods show promise as effective and reliable methods for higher dimensional adaptive grid generation

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

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

sections of convex bodies with symmetries abstract in this paper we study how certain symmetries of convex bodies affect their geometric properties in particular we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations generalizing complex and quaternionic convex bodies we conduct a systematic study of sections of bodies with symmetries of this type with the emphasis on problems of the busemann petty type and hyperplane inequalities the main role belongs to the class of intersection bodies with symmetries

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

intersection bodies in r4 intersection bodies were introduced in 1988 by lutwak who found a close connection between those bodies and the well known 1956 busemann petty problem which asks whether origin symmetric convex bodies with larger central hyperplane sections also have greater volume the author has recently shown that an origin symmetric star bodykis an intersection body if and only if the function x 1kis a positive definite distribution where x k min a 0 x ak we use this result to prove that the unit balls of the spaces l4q 2 q are intersection bodies forn 5 the unit balls of the spaces lnq 2 q are not intersection bodies the technique of this paper allows to find precise expressions for the generating measures signed measures of these bodies the result that the unit cube in r4is an intersection body shows that the negative solution to the busemann petty problem forn 4 in 20 was incorrect

the busemann petty problem via spherical harmonics abstract the busemann petty problem asks whether symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume the solution has recently been completed and the answer is affirmative if n 4 and negative if n 5 in this article we present a short proof of the affirmative result and its generalization using the funk hecke formula for spherical harmonics

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

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

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

an example related to the slicing inequality for general measures abstract for n n let s n be the smallest number s 0 satisfying the inequality k f s k 1 n max s n 1 k f for all centrally symmetric convex bodies k in r n and all even continuous probability densities f on k here k is the volume of k it was proved in 16 that s n 2 n and in analogy with bourgain s slicing problem it was asked whether s n is bounded from above by a universal constant in this note we construct an example showing that s n c n log log n where c 0 is an absolute constant additionally for any 0 2 we describe a related example that satisfies the so called condition

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

a finite volume scheme for the patlak keller segel chemotaxis model a finite volume method is presented to discretize the patlak keller segel pks modeling chemosensitive movements first we prove existence and uniqueness of a numerical solution to the proposed scheme then we give a priori estimates and establish a threshold on the initial mass for which we show that the numerical approximation converges to the solution to the pks system when the initial mass is lower than this threshold numerical simulations are performed to verify accuracy and the properties of the scheme finally in the last section we investigate blow up of the solution for large mass

positive definite distributions and normed spaces we answer a question of alex koldobsky we show that for each p 2 and each n 3 p there is a normed space x of dimension n which embeds in ls if and only if n s p

generalized intersection bodies are not equivalent abstract in a koldobsky a functional analytic approach to intersection bodies geom funct anal 10 2000 1507 1526 a koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent the structures of these two types of generalized intersection bodies have been studied by the author in e milman generalized intersection bodies j funct anal 240 2 2006 530 567 providing substantial evidence for a positive answer to this question the purpose of this note is to construct a counter example which provides a surprising negative answer to this question in a strong sense this implies the existence of non trivial non negative functions in the range of the spherical radon transform and the existence of non trivial spaces which embed in l p for certain negative values of p

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

on dual quermassintegrals of mixed intersection bodies in this paper it is shown that a family of inequalities involving mixed intersection bodies holds the busemann intersection inequality is the first of this family all of the members of this family are strengthened versions of classical inequalities between pairs of dual quermassintegrals of a star body

adaptive grid modelling for cancer cells in the early stage of invasion modelling the interaction of the urokinase plasminogen activator system with a model for cancer cells dynamics lead to a system of five coupled partial differential equations the solution of which over a one dimensional domain can be obtained using the finite element method as the discretization of the integration domain is crucial particularly in the presence of strong variations of the domain variables we tackle the problem solution implementing an adaptive grid numerical technique our results improve previous numerical simulations performed on uniformly discretized domains allowing to capture finer spatial details characterizing the interface between cancer and healthy cells which can be related to malignancy low values of the cancer cells motility induce a high spatial heterogeneity at the cancer healthy cells interface while the dynamical evolution over the whole domain shows branched spatio temporal patterns on the contrary higher motility values smooth the cancer cells spatial profile letting cancer cell concentration to evolve according to a less heterogeneous pattern

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

precise computations of chemotactic collapse using moving mesh methods we consider the problem of computing blow up solutions of chemotaxis systems or the so called chemotactic collapse in two spatial dimensions such solutions can have approximate self similar behaviour which can be very challenging to verify in numerical simulations cf betterton and brenner collapsing bacterial cylinders phys rev e 64 2001 061904 we analyse a dynamic scale invariant remeshing method which performs spatial mesh movement based upon equidistribution using a suitably chosen monitor function the numerical solution resolves the fine detail in the asymptotic solution structure such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by herrero and velazquez singularity patterns in a chemotaxis model math ann 306 1996 583 623 we believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected

size dependent standard deviation for growth rates empirical results and theoretical modeling we study annual logarithmic growth rates r of various economic variables such as exports imports and foreign debt for each of these variables we find that the distributions of r can be approximated by double exponential laplace distributions in the central parts and power law distributions in the tails for each of these variables we further find a power law dependence of the standard deviation sigma r on the average size of the economic variable with a scaling exponent surprisingly close to that found for the gross domestic product gdp phys rev lett 81 3275 1998 by analyzing annual logarithmic growth rates r of wages of 161 different occupations we find a power law dependence of the standard deviation sigma r on the average value of the wages with a scaling exponent beta 0 14 close to those found for the growth of exports imports debt and the growth of the gdp in contrast to these findings we observe for payroll data collected from 50 states of the usa that the standard deviation sigma r of the annual logarithmic growth rate r increases monotonically with the average value of payroll however also in this case we observe a power law dependence of sigma r on the average payroll with a scaling exponent beta 0 08 based on these observations we propose a stochastic process for multiple cross correlated variables where for each variable i the distribution of logarithmic growth rates decays exponentially in the central part ii the distribution of the logarithmic growth rate decays algebraically in the far tails and iii podobnik the standard deviation of the logarithmic growth rate depends algebraically on the average size of the stochastic variable

stability in the busemann petty and shephard problems abstract a comparison problem for volumes of convex bodies asks whether inequalities f k f l for all s n 1 imply that vol n k vol n l where k l are convex bodies in r n and f k is a certain geometric characteristic of k by linear stability in comparison problems we mean that there exists a constant c such that for every e 0 the inequalities f k f l e for all s n 1 imply that vol n k n 1 n vol n l n 1 n c e we prove such results in the settings of the busemann petty and shephard problems and their generalizations we consider the section function f k s k vol n 1 k and the projection function f k p k vol n 1 k where is the central hyperplane perpendicular to and k is the orthogonal projection of k to in these two cases we prove linear stability under additional conditions that k is an intersection body or l is a projection body respectively then we consider other functions f k which allow to remove the additional conditions on the bodies in higher dimensions

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 complex busemann petty problem on sections of convex bodies abstract the complex busemann petty problem asks whether origin symmetric convex bodies in c n with smaller central hyperplane sections necessarily have smaller volume we prove that the answer is affirmative if n 3 and negative if n 4

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

intersection bodies that are not polar zonoids a flat top condition in dimensions four and six abstract we find general geometric conditions on a convex body of revolution k in dimensions four and six so that its intersection body i k is not a polar zonoid we exhibit several examples of intersection bodies which are not polar zonoids

volume bounds for shadow covering for n 2 a construction is given for a large family of compact convex sets k and l in rn such that the orthogonal projection lu onto the subspace u contains a translate of the corresponding projection ku for every direction u while the volumes of k and l satisfy vn k vn l it is subsequently shown that if the orthogonal projection lu onto the subspace u contains a translate of ku for every direction u then the set n n 1 l contains a translate of k it follows that vn k n n 1 n vn l in particular we derive a universal constant bound vn k 2 942 vn l independent of the dimension n of the ambient space related results are obtained for projections onto subspaces of some fixed intermediate co dimension open questions and conjectures are also posed

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

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

a n estimate for measures of hyperplane sections of convex bodies the hyperplane or slicing problem asks whether there exists an absolute constant c so that for any origin symmetric convex body k in rn k n 1n cmax sn 1 k where is the central hyperplane in rn perpendicular to and k stands for volume of proper dimension the problem is still open with the best to date estimate c n1 4 established by klartag who slightly improved the previous estimate of bourgain it is much easier to get a weaker estimate with c n in this note we show that the n estimate holds for arbitrary measure in place of volume namely if l is an origin symmetric convex body in rn and is a measure with non negative even continuous density on l then l nnn 1cnmax sn 1 l l 1 n where cn b2n n 1n b2n 1 1 and b2n is the unit euclidean ball in rn we deduce this inequality from a stability result for intersection bodies

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

a positivity preserving adaptive moving mesh method for cancer cell invasion models abstract in this paper we present an efficient adaptive moving mesh finite difference method for the modeling of early stage cancer cell invasion of tissue or extracellular matrix ecm the cancer cell invasion model is a nonlinear system of reaction diffusion taxis partial differential equations pdes describing the time evolution of the cancer cells density and concentrations of proteins of the ecm the solution of the model exhibits very rapid variations at the boundary of the healthy and cancer cells thus using a uniform grid method to solve the cancer cell invasion model requires a very large number of grid points in order to resolve the steep gradients of the solution precisely as a result the computation can become prohibitively expensive and inefficient in this effort we propose a positivity preserving scheme for the spatial discretization of the model on an adaptive mesh to accurately capture the sharp structures of the solution the adaptive mesh is generated by a coordinate transformation obtained as the solution of the optimal mass transfer problem several numerical experiments are presented to demonstrate the performance of the adaptive mesh method for solving the cancer cell invasion model the numerical results show that the proposed adaptive mesh method is effective in capturing and predicting the correct behavior of the cancer cells movement within the ecm

gaussian measure of sections of convex bodies abstract in this paper we study properties of sections of convex bodies with respect to the gaussian measure we develop a formula connecting the minkowski functional of a convex symmetric body k with the gaussian measure of its sections using this formula we solve an analog of the busemann petty problem for gaussian measures

on polars of mixed projection bodies abstract recently lutwak established general minkowski inequality brunn minkowski inequality and aleksandrov fenchel inequality for mixed projection bodies in this paper following lutwak we established their polar forms as applications we prove some interrelated results

the growth of business firms theoretical framework and empirical evidence we introduce a model of proportional growth to explain the distribution pg g of business firm growth rates the model predicts that pg g is exponential in the central part and depicts an asymptotic power law behavior in the tails with an exponent 3 because of data limitations previous studies in this field have been focusing exclusively on the laplace shape of the body of the distribution in this article we test the model at different levels of aggregation in the economy from products to firms to countries and we find that the predictions of the model agree with empirical growth distributions and size variance relationships

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

intersection bodies and lp spaces abstract we prove that the convex intersection bodies are isomorphically equivalent to unit balls of subspaces of l q for each q 0 1 this is done by extending to negative values of p the factorization theorem of maurey and nikishin which states that for any 0 p q 1 every banach subspace of l p is isomorphic to a subspace of l q

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

second derivative test for intersection bodies in 1956 busemann and petty asked whether symmetric convex bodies in rnwith larger central hyperplane sections also have greater volume this question was answered in the negative forn 5 in a series of papers giving individual counterexamples in 1988 lutwak introduced the concept of an intersection body and proved that every smooth nonintersection body in rnprovides a counterexample to the busemann petty problem in this article we use the connection between intersection bodies and positive definite distributions established by the author in an earlier paper to give a necessary condition for intersection bodies in terms of the second derivative of the norm this result allows us to produce a variety of counterexamples to the busemann petty problem in rn n 5 for example the unit ball of theq sum of any finite dimensional normed spacesxandywithq 2 dim x 1 dim y 4 is not an intersection body as well as the unit balls of the orlicz spaces lnm n 5 withm 0 m 0 0

the busemann petty problem for arbitrary measures the busemann petty problem asks whether symmetric convex bodies in n with smaller n 1 dimensional volume of central hyperplane sections necessarily have smaller n dimensional volume the answer to this problem is affirmative for n 4 and negative for n 5 in this paper we generalize the busemann petty problem to essentially arbitrary measure in place of the volume we also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in n

relations between allometric scalings and fluctuations in complex systems the case of japanese firms abstract to elucidate allometric scaling in complex systems we investigated the underlying scaling relationships between typical three scale indicators for approximately 500 000 japanese firms namely annual sales number of employees and number of business partners first new scaling relations including the distributions of fluctuations were discovered by systematically analyzing conditional statistics second we introduced simple probabilistic models that reproduce all these scaling relations and we derived relations between scaling exponents and the magnitude of fluctuations

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

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

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