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We study extremal properties of spherical random polytopes, the convex hull of random points chosen from the unit Euclidean sphere in $\mathbb{R}^n$. The extremal properties of interest are the expected values of the maximum and minimum surface area among facets. We determine the asymptotic growth in every fixed dimension, up to absolute constants.

We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santal\'o inequality and the $L_p$-affine isoperimetric inequalities, for convex bodies in spherical space. Specifically, we establish an isoperimetric inequality for the floating area and prove a stability result based on the spherical volume difference. The floating area has previously been studied as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant curvature. In this work, we introduce the $L_p$-floating areas for spherical convex bodies, extending Lutwak's centro-affine invariant family of $L_p$-affine surface area measures from Euclidean geometry. We prove a duality formula, monotonicity properties, and isoperimetric inequalities associated with this new family of curvature measures for spherical convex bodies. Additionally, we propose a novel curvature entropy functional for spherical convex bodies, based on the $L_p$-floating area, and establish a corresponding dual isoperimetric inequality. Finally, we extend our spherical notions to space forms with non-negative constant curvature in two distinct ways. One extension asymptotically connects with centro-affine geometry on convex bodies as curvature approaches zero, while the other converges with Euclidean geometry. Notably, our newly introduced curvature entropy for spherical convex bodies emerges as a natural counterpart to both the centro-affine entropy and the Gaussian entropy of convex bodies in Euclidean space., Comment: 43 pages

While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant $K_n$ is given by the convex hull of $n$ independent random points chosen uniformly on the boundary or in the interior of $K$. When $K$ is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance between a polygon as $n\to\infty$. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices., Comment: 17 pages, 2 figures

The purpose of this paper is to introduce the new concept of weighted floating functions associated with log concave or $s$-concave functions. This leads to new notions of weighted functional affine surface areas. Their relation to more traditional versions of functional affine surface areas as well as to the classical affine surface areas for convex bodies is discussed in detail., Comment: arXiv admin note: text overlap with arXiv:1711.11088

We introduce extremal affine surface areas in a functional setting. We show their main properties. Among them are linear invariance, isoperimetric inequalities and monotonicity properties. We establish a new duality formula, which shows that the maximal (resp. minimal) inner affine surface area of an $s$-concave function on $\mathbb{R}^n$ equals the maximal (resp. minimal) outer affine surface area of its Legendre polar. We estimate the ``size" of these quantities: up to a constant depending on $n$ and $s$ only, the extremal affine surface areas are proportional to a power of the integral of $f$. This extends results obtained in the setting of convex bodies. We recover and improve those as a corollary to our results.

We investigate a natural analog to Lutwak's $p$-affine surface area in $d$-dimensional spherical, hyperbolic and de Sitter space. In particular, we show that these curvature measures appear naturally as the volume derivative of floating bodies of non-Euclidean convex bodies conjugated by duality, such as spherical, hyperbolic and de Sitter convex bodies. We provide a unifying framework by establishing a real-analytic version of this relation controlled by the constant curvature of the $d$-dimensional real space form. These new curvature measures relate in two distinctly different ways to curvature measures on Euclidean space, one of which is Lutwak's centro-affine invariant $p$-affine surface area, and the other is related to a rigid-motion invariant curvature measure that appears naturally as the volume derivative of Schneider's mean-width separation body., Comment: 40 pages, 9 figures

We investigate the weighted $L_p$ affine surface areas which appear in the recently established $L_p$ Steiner formula of the $L_p$ Brunn Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. We show that they are related to $f$ divergences of the cone measures of the convex body and its polar, namely the Kullback-Leibler divergence and the R\'enyi-divergence.

We give an overview of the affine surface area, its properties and its history.

We extend the notion of Ulam floating sets from convex bodies to Ulam floating functions. We use the Ulam floating functions to derive a new variational formula for the affine surface area of log-concave functions.

Using a natural representation of a $1/s$-concave function on $\mathbb{R}^d$ as a convex set in $\mathbb{R}^{d+1},$ we derive a simple formula for the integral of its $s$-polar. This leads to convexity properties of the integral of the $s$-polar function with respect to the center of polarity. In particular, we prove that that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santal\'o regions for $s$-concave and log-concave functions and generalize the Santal\'o inequality for them in the case the origin is not the Santal\'o point.

In this paper, we give an overview of some results concerning best and random approximation of convex bodies by polytopes. We explain how both are linked and see that random approximation is almost as good as best approximation., Comment: 22 pages

Inspired by an $L_p$ Steiner formula for the $L_p$ affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, $L_p$-Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the $L_p$ affine surface areas and the mixed $L_p$ affine surface areas. We investigate the properties of the $L_p$-Steiner quermassintegrals in a special class of convex bodies. In particular, we show that they are rotation and reflection invariant valuations in this class of convex bodies with a certain degree of homogeneity. Such valuations seem new and have not been observed before.

Affine invariant points and maps for sets were introduced by Gr\"unbaum to study the symmetry structure of convex sets. We extend these notions to a functional setting. The role of symmetry of the set is now taken by evenness of the function. We show that among the examples for affine invariant points are the classical center of gravity of a log-concave function and its Santal\'o point. We also show that the recently introduced floating functions and the John- and L\"owner functions are examples of affine invariant maps. Their centers provide new examples of affine invariant points for log-concave functions.

Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a "pointwise Prekopa-Leindler inequality" and show a monotonicity property of the multimarginal Blaschke-Santalo functional.

In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.

Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is established also for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $n$-dimensional $\ell_p$-ball. In particular, this includes the cone and the uniform probability measure.

We introduce the notion of Loewner (ellipsoid) function for a log concave function and show that it is an extension of the Loewner ellipsoid for convex bodies. We investigate its duality relation to the recently defined John (ellipsoid) function by Alonso-Gutierrez, Merino, Jimenez and Villa. For convex bodies, John and Loewner ellipsoids are dual to each other. Interestingly, this need not be the case for the John function and the Loewner function.

We prove an analogue of the classical Steiner formula for the $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory and the dual quermassintegrals of the dual Brunn Minkowski theory as well as special cases. Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the $s$-th mixed $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$ and~$s$ is also given.

The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of the involved sets, more precisely, by what is called the surface area deviation. The proof uses arguments and constructions from probability, convex and integral geometry. The bound is closely related to $p$-affine surface areas.

Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.

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.

We study a new construction of bodies from a given convex body in $\mathbb{R}^{n}$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas. We show that these bodies are related to Ulam's long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation., Comment: 25 pages, 3 figures

We introduce floating bodies for convex, not necessarily bounded subsets of $\mathbb{R}^n$. This allows us to define floating functions for convex and log concave functions and log concave measures. We establish the asymptotic behavior of the integral difference of a log concave function and its floating function. This gives rise to a new affine invariant which bears striking similarities to the Euclidean affine surface area.

We consider the question how well a floating body can be approximated by the polar of the illumination body of the polar. We establish precise convergence results in the case of centrally symmetric polytopes. This leads to a new affine invariant which is related to the cone measure of the polytope., Comment: arXiv admin note: text overlap with arXiv:1709.02424

We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question for the class of centrally symmetric convex bodies. We provide precise estimates for $(B_p^n)$ and for centrally symmetric convex bodies with everywhere positive Gauss curvature. Our estimates show that equality of the polar of the floating body and the illumination body of the polar can only be achieved in the case of ellipsoids.

Let $K$ be a convex body in $\mathbb{R}^n$ and $f : \partial K \rightarrow \mathbb{R}_+$ a continuous, strictly positive function with $\int\limits_{\partial K} f(x) d \mu_{\partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmetric difference metric by an arbitrarily positioned polytope $P_f$ in $\mathbb{R}^n$ having a fixed number of vertices. This generalizes a result by Ludwig, Sch\"utt and Werner $[36]$. The polytope $P_f$ is obtained by a random construction via a probability measure with density $f$. In our result, the dependence on the number of vertices is optimal. With the optimal density $f$, the dependence on $K$ in our result is also optimal.

Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floating areas are combined to derive new asymptotic approximation results on the sphere, in hyperbolic space and in Hilbert geometries.

In this paper, the concept of the classical $f$-divergence for a pair of measures is extended to the mixed $f$-divergence for multiple pairs of measures. The mixed $f$-divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed $f$-divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov-Fenchel type inequality and an isoperimetric inequality for the mixed $f$-divergence are proved., Comment: arXiv admin note: substantial text overlap with arXiv:1304.6792

We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies.

While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices in the symmetric surface area deviation.

Mixed $f$-divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov-Fenchel type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log concave functions. Special cases of $f$-divergences are mixed $L_\lambda$-affine surface areas for log concave functions. For those, we establish various affine isoperimetric inequalities as well as a vector Blaschke Santal\'{o} type inequality.

The Blaschke Santal\'o inequality and the $L_p$ affine isoperimetric inequalities are major inequalities in convex geometry and they have a wide range of applications. Functional versions of the Blaschke Santal\'o inequality have been established over the years through many contributions. More recently and ongoing, such functional versions have been established for the $L_p$ affine isoperimetric inequalities as well. These functional versions involve notions from information theory, like entropy and divergence. We list stability versions for the geometric inequalities as well as for their functional counterparts. Both are known for the Blaschke Santal\'o inequality. Stability versions for the $L_p$ affine isoperimetric inequalities in the case of convex bodies have only been known in all dimensions for $p=1$ and for $p > 1$ only for convex bodies in the plane. Here, we prove almost optimal stability results for the $L_p$ affine isoperimetric inequalities, for all $p$, for all convex bodies, for all dimensions. Moreover, we give stability versions for the corresponding functional versions of the $L_p$ affine isoperimetric inequalities, namely the reverse log Sobolev inequality, the $L_p$ affine isoperimetric inequalities for log concave functions and certain divergence inequalities.

An affine invariant point on the class of convex bodies in R^n, endowed with the Hausdorff metric, is a continuous map p which is invariant under one-to-one affine transformations A on R^n, that is, p(A(K))=A(p(K)). We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K^{p(K)})=p(K) for every convex body K, where K^{p(K)} denotes the polar of K with respect to p(K). We investigate which affine invariant points do have a dual point, whether this dual point is unique and has itself a dual point. We define a product on the set of affine invariant points, in relation with duality. Finally, examples are given which exhibit the rich structure of the set of affine invariant points.

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.

In this paper, the concept of the classical $f$-divergence (for a pair of measures) is extended to the mixed $f$-divergence (for multiple pairs of measures). The mixed $f$-divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed $f$-divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov-Fenchel type inequality and an isoperimetric type inequality for the mixed $f$-divergence will be proved and applications in the theory of convex bodies are given.

We answer in the negative a question by Gruenbaum who asked if there exists a finite basis of affine invariant points. We give a positive answer to another question by Gruenbaum about the "size" of the set of all affine invariant points. Related, we show that the set of all convex bodies K, for which the set of affine invariant points is all of n-dimensional Euclidean space, is dense in the set of convex bodies. Crucial to establish these results, are new affine invariant points, not previously considered in the literature.

We introduce f-divergence, a concept from information theory and statistics, for convex bodies in R^n. We prove that f-divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the L_p affine surface area from the L_p Brunn Minkowski theory are special cases of f-divergences.

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1, lim_{p\rightarrow \infty} \frac{p}{\log{p}} (\frac{|Z_{p}^{\circ}(P)|}{|P^{\circ}|} -1) = n^{2}. We provide an application to the approximation of polytopes by uniformly convex sets.

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 R\'enyi divergences of the cone measures of a convex body and its polar. We give geometric interpretations for all R\'enyi divergences $D_\alpha$, not just for the previously treated special case of relative entropy which is the case $\alpha =1$. 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 makes them appear. Thus, in the new approach we detect "faster" details about the boundary of a convex body.

Let $K$ be a convex body in $\mathbb{R}^n$ with Santal\'o point at 0\. We show that if $K$ has a point on the boundary with positive generalized Gau{\ss} curvature, then the volume product $|K| |K^\circ|$ is not minimal. This means that a body with minimal volume product has Gau{\ss} curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.

We give an example of a convex body whose centroid and Santal\'o point are "far apart".

Let $K$ be a convex body in $\bbR^n$ and $\d>0$. The homothety conjecture asks: Does $K_{\d}=c K$ imply that $K$ is an ellipsoid? Here $K_{\d}$ is the (convex) floating body and $c$ is a constant depending on $\d$ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies $B^n_p$, $1\leq p\leq \infty$, the unit balls of $l_p^n$; namely, we show that $(B^n_p)_{\d} = c B^n_p$ if and only if $p=2$. We also show that the homothety conjecture is true for a general convex body $K$ if $\d$ is small enough. This improvs earlier results by Sch\"utt and Werner \cite{SW1994} and Stancu \cite{Stancu2009}., Comment: 24 pages, 2 figures

Let $K$ be a convex body in $\mathbb R^n$. We introduce a new affine invariant, which we call $\Omega_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone measure of $K$ and the cone measure of $K^\circ$, as the limit of the volume difference of $K$ and $L_p$-centroid bodies. We investigate properties of $\Omega_K$ and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show a "information inequality" for convex bodies.

We show that a bipartite state on a tensor product of two matrix algebras is almost surely entangled if its rank is not greater than that of one of its reduced density matrices.