general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

crame spl acute r rao and moment entropy inequalities for renyi entropy and generalized fisher information the moment entropy inequality shows that a continuous random variable with given second moment and maximal shannon entropy must be gaussian stam s inequality shows that a continuous random variable with given fisher information and minimal shannon entropy must also be gaussian the crame spl acute r rao inequality is a direct consequence of these two inequalities in this paper the inequalities above are extended to renyi entropy p sup th moment and generalized fisher information generalized gaussian random densities are introduced and shown to be the extremal densities for the new inequalities an extension of the crame spl acute r rao inequality is derived as a consequence of these moment and fisher information inequalities

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

a characterization of affine surface area abstract we show that every upper semicontinuous and equi affine invariant valuation on the space of d dimensional convex bodies is a linear combination of affine surface area volume and the euler characteristic

curvature flows in the sphere we consider contracting and expanding curvature flows in mathbb s n 1 when the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the gaus map the contracting hypersurfaces shrink to a point x 0 while the expanding hypersurfaces converge to the equator of the hemisphere mathcal h x 0 after rescaling by the same scale factor the rescaled hypersurfaces converge to the unit spheres with centers x 0 resp x 0 exponentially fast in c infty mathbb s n

the discrete planar l0 minkowski problem abstract in the discrete setting the l 0 minkowski problem extends the question posed and answered by the classical minkowski s existence theorem for polytopes in particular the planar extension which we address in this paper concerns the existence of a convex polygonal body which contains the origin whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area

what does finite volume based implicit filtering really resolve in large eddy simulations the study illustrated in this paper completes the topics initially investigated in ref 42 the aim being here to analyze the role of the integral based finite volume fv discretizations in large eddy simulations that exploit the implicit filtering approach specifically a theoretical study on the effective shape and length of three dimensional filters induced by some fv based flux reconstructions is the object of this paper for any integral based flux reconstruction one gets always an approximation of the top hat filter kernel this is not the case of the filters induced by the differential based finite difference operators such as those reported and analyzed in refs 16 24 considering the sub filter resolution parameter email protected e f f h being d e f f the effective filter width and h the computational grid size allows us discerning the effective measure of the approximate built in top hat filter the induced shape and width is analyzed by means of a modified wavenumber like analysis that is developed in the 3d fourier space several evaluation criteria applied on different schemes are considered and the differences in terms of either velocity or flux interpolations on staggered or non staggered grids are analyzed conclusions are reported that depending on the using of either the integral or the differential form of the filtered equations the induced numerical filter is or is not a congruent approximation of the exact top hat transfer function for some value q the need of a suitable estimation of the sub filter parameter q is assessed from several real les computations that make use of the new integral based version of the eddy viscosity dynamic modeling presented in ref 42 in fact it is shown that the test filtering length has to be carefully chosen as a function of the fv based induced filter

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

constrained convex bodies with extremal affine surface areas abstract given a convex body k r n and p r we introduce and study the extremal inner and outer affine surface areas i s p k sup k k as p k and o s p k inf k k as p k where as p k denotes the l p affine surface area of k and the supremum is taken over all convex subsets of k and the infimum over all convex compact subsets containing k the convex body that realizes i s 1 k in dimension 2 was determined in 3 where it was also shown that this body is the limit shape of lattice polytopes in k in higher dimensions no results are known about the extremal bodies we use a thin shell estimate of 23 and the lowner ellipsoid to give asymptotic estimates on the size of i s p k and o s p k surprisingly it turns out that both quantities are proportional to a power of volume

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

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

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 spherical convex floating body abstract for a convex body on the euclidean unit sphere the spherical convex floating body is introduced the asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated this gives rise to a new spherical area measure the floating area remarkably this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry several properties of the floating area are established

on the number of solutions to the discrete two dimensional l0 minkowski problem abstract our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices if they exist such polygons are solutions to the discrete l 0 minkowski problem in the plane proposed in a larger generality by lutwak in his extension of the brunn minkowski theory the method exploits the equivalence between the l 0 polygons and the homothetic solutions to appropriate anisotropic crystalline flows hence as a by product we obtain a positive answer to jean taylor s conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature without any symmetry assumptions

contravariant minkowski valuations a complete classification of all continuous gl n contravariant minkowski valuations is established as an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical petty projection inequality

embedding sn into rn 1 with given integral gauss curvature and optimal mass transport on sn in a d aleksandrov convex polyhedra gittl moscow ussr 1950 in russian english translation a d aleksandrov convex polyhedra springer berlin new york 2005 a d aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data the first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in euclidean space with prescribed integral gauss curvature our solution includes the case of a convex polytope this problem was also first considered by aleksandrov and below it is referred to as aleksandrov s problem the second goal of this paper is to show that in variational form the aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations

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

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

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

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

resolving the singularities in the space of riemannian geometries a method is described for unfolding the singularities of superspace g m d the space of riemannian geometries of a manifold m this extended or unfolded superspace is described by the projection gf m m f m d m d g where f m is the frame bundle of m the unfolded space gf m is an infinite dimensional manifold without singularities moreover as expected the unfolding of gf m at each geometry g0 g is parametrized by the isometry group ig0 m of g0 the construction is completely natural gives complete control and knowledge of the unfolding at each geometry necessary to make gf m a manifold and is generally covariant with respect to all coordinate transformations a similar program is outlined based on the methods of this paper of desingularizing the moduli space of connections on a principal fiber bundle

extended affine surface area during the twentieth century many of the notions of classical differential geometry have been extended so as to be defined on the boundaries of arbitrary convex bodies without smoothness requirements for example the projection measures quermassintegrals are the natural extensions of the integrals of mean curvature see santa16 27 pp 215 2321 for a discussion the surface area measures of aleksandrov fenchel jessen are extensions of the indefinite integrals of the elementary symmetric functions of the principal radii of curvature while federer s curvature measures when restricted to convex hypersurfaces are extensions of the indefinite integrals of the elementary symmetric functions of the principal curvatures see schneider 30 pp 27 331 one of the important concepts of classical differential geometry which has been difficult to extend is afline surface area this has often led to difficulties which when circumvented have been done so only by considerable effort and ingenuity for example suppose we are given a measure on the unit sphere which satisfies the hypothesis of the minkowski problem the solution of the minkowski problem guarantees the existence of a convex body whose surface area measure is the given measure from the given measure we can find the surface area of the body the minkowski mixed volume inequality can be used to obtain various upper bounds for the volume of the body however we cannot even inquire about the affine surface area of the body unless we can somehow convince ourselves that the convex body is sufficiently smooth results regarding aftine surface area have a surprising number of applications for example a number of geometric inequalities which appear to have no connection with affine surface area are in fact consequences of inequalities involving afline surface area see for example petty 25 and also 151 since polytopes have zero afftne surface area it seems quite surprising that the notion of afline surface area arises frequently in the area of polytope approximation see the survey of gruber 9 recently leichtweiss 11 121 presented a definition of extended afftne

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

integral geometry and the gauss bonnet theorem in constant curvature spaces we give an integral geometric proof of the gauss bonnet theorem for hypersurfaces in constant curvature spaces as a tool we obtain variation formulas in integral geometry with interest in its own

cones generated by random points on half spheres and convex hulls of poisson point processes let u 1 u 2 ldots be random points sampled uniformly and independently from the d dimensional upper half sphere we show that as n rightarrow infty the f vector of the d 1 dimensional convex cone c n generated by u 1 ldots u n weakly converges to a certain limiting random vector without any normalization we also show convergence of all moments of the f vector of c n and identify the limiting constants for the expectations we prove that the expected grassmann angles of c n can be expressed through the expected f vector this yields convergence of expected grassmann angles and conic intrinsic volumes and answers thereby a question of barany et al random struct algorithms 50 1 3 22 2017 https doi org 10 1002 rsa 20644 our approach is based on the observation that the random cone c n weakly converges after a suitable rescaling to a random cone whose intersection with the tangent hyperplane of the half sphere at its north pole is the convex hull of the poisson point process with power law intensity function proportional to vert x vert d gamma where gamma 1 we compute the expected number of facets the expected intrinsic volumes and the expected t functional of this random convex hull for arbitrary gamma 0

functional affine isoperimetry and an inverse logarithmic sobolev inequality abstract we give a functional version of the affine isoperimetric inequality for log concave functions which may be interpreted as an inverse form of a logarithmic sobolev inequality for entropy a linearization of this inequality gives an inverse inequality to the poincare inequality for the gaussian measure

relative entropy of cone measures and lp centroid bodies let k be a convex body in r n we introduce a new affine invariant which we call k t hat can be found in three different ways a as a limit of normalized lp affine surface areas b as the relative entropy of the cone measure of k and the cone measure of k c as the limit of the volume difference of k and lp centroid bodies we investigate properties of k and of related new invariant quantities in particular we show new affine isoperimetric inequalities and we show an information inequality for convex bodies

monotonicity of facet numbers of random convex hulls abstract let x 1 x n be independent random points that are distributed according to a probability measure on r d and let p n be the random convex hull generated by x 1 x n n d 1 for natural classes of probability distributions and by means of blaschke petkantschin formulae from integral geometry it is shown that the mean facet number of p n is strictly monotonically increasing in n

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

volume approximation of convex bodies by inscribed polytopes a convex body c in d dimensional euclidean space e is a compact convex subset of le a with non empty interior we say that c is of differentiability class k if its boundary considered as a manifold is of class k the symmetric difference metric 6 s also called nikodym metric on the space of convex bodies is defined for any two convex bodies as the volume of their symmetric difference given a convex body c let c n d l d 2 denote the family of convex polytopes p having at most n vertices and being inscribed into c that is all vertices are on the boundary bdc of c it is well known that for each n there is a polytope p such that s c p s c inf s c p p

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

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

the affine plateau problem in this paper we study a second order variational problem for locally convex hypersurfaces which is the affine invariant analogue of the classical plateau problem for minimal surfaces we prove existence regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions