laplace transforms and valuations it is proved that the classical laplace transform is a continuous valuation which is positively gl n covariant and logarithmic translation covariant conversely these properties turn out to be sufficient to characterize this transform

on extendability by continuity of valuations on convex polytopes abstract there is a well known construction of weakly continuous valuations on convex compact polytopes in r n in this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the hausdorff metric to all convex compact subsets of r n it is shown that there is a necessary condition on the initial data for such an extension in the case of r 3 more explicit results are obtained

valuations on sobolev spaces all affinely covariant convex body valued valuations on the sobolev space w 1 1 bbb r n are completely classified it is shown that there is a unique such valuation for blaschke addition this valuation turns out to be the operator which associates with each function f in w 1 1 bbb r n the unit ball of its optimal sobolev norm

inequalities for mixed p affine surface area we prove new alexandrov fenchel type inequalities and new affine isoperimetric inequalities for mixed p affine surface areas we introduce a new class of bodies the illumination surface bodies and establish some of their properties we show for instance that they are not necessarily convex we give geometric interpretations of l p affine surface areas mixed p affine surface areas and other functionals via these bodies the surprising new element is that not necessarily convex bodies provide the tool for these interpretations

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

lp dual affine surface area abstract according to the notion of l p affine surface area by lutwak in this paper we introduce the concept of l p dual affine surface area further we establish the affine isoperimetric inequality and the blaschke santalo inequality for l p dual affine surface area besides the dual brunn minkowski inequality for l p dual affine surface area is presented

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

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

new orlicz affine isoperimetric inequalities abstract the orlicz brunn minkowski theory received considerable attention recently and many results in the l p brunn minkowski theory have been extended to their orlicz counterparts the aim of this paper is to develop orlicz l affine and geominimal surface areas for a single convex body as well as for multiple convex bodies which generalize the l p mixed affine and geominimal surface areas fundamental concepts in the l p brunn minkowski theory our extensions are different from the general affine surface areas by ludwig in 21 moreover our definitions for orlicz l affine and geominimal surface areas reveal that these affine invariants are essentially the infimum supremum of v k l the orlicz mixed volume of k and the polar body of l where l runs over all star bodies and all convex bodies respectively with volume of l equal to the volume of the unit euclidean ball b 2 n properties for the orlicz l affine and geominimal surface areas such as affine invariance and monotonicity are proved related orlicz affine isoperimetric inequalities are also established

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 particle algorithm for sequential bayesian parameter estimation and model selection we describe a particle algorithm for the sequential bayesian estimation of unknown static parameters the algorithm combines sequential importance sampling sis and markov chain monte carlo mcmc to achieve computational efficiency and stability in its most general form the algorithm has three components i sis ii a rejuvenation test and iii mcmc measurements are processed sequentially with an artificial time line if there is no natural one associated with the measurements by sis which is computationally inexpensive after each measurement is processed the rejuvenation test checks whether the resulting sis particles have to be rejuvenated when indicated by the test the particles are completely rejuvenated by mcmc which removes errors that accumulate from sis due to the finite number of particles thus ensuring stability whenever possible the sis particles can be used to advantage in the mcmc there is flexibility in the choice of the rejuvenation test as well as the mcmc method with potential to increase the usefulness of the algorithm in particular by using reversible jump mcnic with multiple models the algorithm can perform simultaneous model selection and parameter estimation in this paper we use a rejuvenation test based on a kullback leibler distance that is easy to compute and our choice of mcnic is independent metropolis hastings with a gaussian proposal density with these choices we illustrate the use of the algorithm in two signal processing applications passive source localization with angle of arrival and simultaneous weak signal detection and parameter estimation involving both simulated and real data the results demonstrate the algorithm s stability its built in protection against model overfitting and tolerance to model mismatch

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

lp minkowski valuations on lq spaces abstract all continuous sl n covariant l p minkowski valuations on l q are classified the only non trivial operators appearing in this context are the asymmetric l p moment operators

on the extendability by continuity of angular valuations on polytopes abstract a classical theorem of p mcmullen describes all valuations on polytopes that are invariant under translations and weakly continuous i e continuous with respect to parallel displacements of the facets of a polytope while it is typically not difficult to check that a valuation is weakly continuous it is not clear how to decide whether it admits a continuous extension to convex bodies in a special case of mcmullen s construction a simple necessary and sufficient condition on the initial data of such an extension is obtained

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

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

difference bodies in complex vector spaces abstract a complete classification is obtained of continuous translation invariant minkowski valuations on an m dimensional complex vector space which are covariant under the complex special linear group

the santalo regions of a convex body motivated by the blaschke santalo inequality we define for a convex body k in r and for t r the santalo regions s k t of k we investigate properties of these sets and relate them to a concept of affine differential geometry the affine surface area of k let k be a convex body in r for x int k the interior of k let k be the polar body of k with respect to x it is well known that there exists a unique x0 int k such that the product of the volumes k k0 is minimal see for instance sch this unique x0 is called the santalo point of k moreover the blaschke santalo inequality says that k k0 v n where vn denotes the volume of the n dimensional euclidean unit ball b 0 1 with equality if and only if k is an ellipsoid for t r we consider here the sets s k t x k k k v2 n t following e lutwak we call s k t a santalo region of k observe that it follows from the blaschke santalo inequality that the santalopoint x0 s k 1 and that s k 1 x0 if and only if k is an ellipsoid thus s k t has non empty interior for some t 1 if and only if k is not an ellipsoid in the first part of this paper we show some properties of s k t and give estimates on the size of s k t this question was asked by e lutwak the paper was written while both authors stayed at msri supported by a grant from the national science foundation msc classification 52

fisher information and matrix valued valuations abstract all continuous and affinely contravariant matrix valued valuations on the sobolev space w 1 2 r n are completely classified it is shown that there is a unique such valuation this valuation turns out to be the fisher information matrix

lp mixed geominimal surface area abstract this paper deals with l p geominimal surface area and its extension to l p mixed geominimal surface area we give an integral formula of l p geominimal surface area by the p petty body and introduce the concept of l p mixed geominimal surface area which is a natural extension of l p geominimal surface area some inequalities such as analogues of alexandrov fenchel inequalities blaschke santalo inequalities and affine isoperimetric inequalities for l p mixed geominimal surface areas are obtained

p q mixed geominimal surface area and p q mixed affine surface area abstract in this paper the p q mixed geominimal surface area and the p q mixed affine surface area are introduced they are extensions of the l p geominimal surface area and the l p affine surface area respectively affine invariance under unimodular affine transformations and continuity for the p q mixed geominimal surface area and the p q mixed affine surface area are discussed the existence the uniqueness and the uniform boundness of the p q mixed petty bodies are obtained affine isoperimetric inequalities for the p q mixed geominimal surface area and the p q mixed affine surface area are established

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

lp mixed affine surface area abstract according to the three notions of mixed affine surface area l p affine surface area and l p mixed affine surface area proposed by lutwak in this article we give the concept of ith l p mixed affine surface area such that the first and second notions of lutwak are its special cases further some lutwak s results are extended associated with this concept besides applying this concept we establish an inequality for the volumes and dual quermassintegrals of a class of star bodies

valuations on polytopes containing the origin in their interiors we give a classification of non negative or borel measurable sl d invariant homogeneous valuations on the space of d dimensional convex polytopes containing the origin in their interiors the only examples are volume volume of the polar body and the euler characteristic

gerby localization z3 hodge integrals and the gw theory of c z3 we exhibit a set of recursive relations that completely determine all equivariant gromov witten invariants of bbb c 3 bbb z 3 we interpret such invariants as bbb z 3 hodge integrals and produce relations among them via atiyah bott localization on moduli spaces of twisted stable maps to gerbes over bbb p 1

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

localization and gluing of topological amplitudes we develop a gluing algorithm for gromov witten invariants of toric calabi yau threefolds based on localization and gluing graphs the main building block of this algorithm is a generating function of cubic hodge integrals of special form we conjecture a precise relation between this generating function and the topological vertex at fractional framing

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

crofton measures and minkowski valuations a description of continuous rigid motion compatible minkowski valuations is established as an application we present a brunn minkowski type inequality for intrinsic volumes of these valuations

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

projection bodies in complex vector spaces the space of minkowski valuations on an m dimensional complex vector space which are continuous translation invariant and contravariant under the complex special linear group is explicitly described each valuation with these properties is shown to satisfy geometric inequalities of the brunn minkowski aleksandrov fenchel and minkowski type

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

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

affine restriction for radial surfaces suppose d is affine surface measure on a convex radial surface x x x a x b in mathbb r 3 under appropriate smoothness and growth conditions on we prove l 4 3 mathbb r 3 l 4 3 d mu and l 4 3 mathbb r 3 l 2 d mu fourier restriction estimates for

valuation theory of indefinite orthogonal groups abstract let so p q denote the identity connected component of the real orthogonal group with signature p q we give a complete description of the spaces of continuous and generalized translation and so p q invariant valuations generalizing hadwiger s classification of euclidean isometry invariant valuations as a result of independent interest we identify within the space of translation invariant valuations the class of klain schneider continuous valuations which strictly contains all continuous translation invariant valuations the operations of pull back and push forward by a linear map extend naturally to this class

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