Affine Surface Area Research Articles

Lp-Steiner quermassintegrals

Inspired by an Lp Steiner formula for the Lp affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, Lp-Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the Lp affine surface areas and the mixed Lp affine surface areas. We investigate the properties of the Lp-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.

The (φ,ψ) Orlicz mixed Petty bodies

In this paper, the homogeneous (φ,ψ) Orlicz mixed affine and geominimal surface areas are introduced. Their homogeneity, affine invariance and affine isoperimetric inequalities are established. Under some assumptions, the existence and uniqueness of the homogeneous (φ,ψ) Orlicz mixed Petty bodies are established. Base on these, the continuity of the homogeneous Orlicz mixed geominimal surface areas is given. Moreover, the nonhomogeneous (φ,ψ) Orlicz mixed affine and geominimal surface areas are defined, and the continuity of the nonhomogeneous (φ,ψ) Orlicz mixed geominimal surface for some special φ and ψ is obtained as well.

Curvature functionals on convex bodies

AbstractWe 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 Rényi divergence.

Floating bodies and approximation of convex bodies by polytopes

We describe some results on approximation of convex bodies by polytopes. Best and random approximations are considered and compared. The geometric quantities related to the convex body, that appear naturally in such approximation questions are the affine surface areas. Those, and their relation to floating bodies, will be discussed as well. In this survey we give only a very selective collection of results on approximation by polytopes.

Pinsker inequalities and related Monge-Ampere equations for log-concave functions

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.

Extremal general affine surface areas

For a convex body K in Rn, we introduce and study the extremal general affine surface areas, defined byISφ(K):=supK′⊂K⁡asφ(K),osψ(K):=infK′⊃K⁡asψ(K) where asφ(K) and asψ(K) are the Lφ and Lψ affine surface area of K, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ISφ and osψ are continuous. In our main results, we prove Blaschke-Santaló type inequalities and inverse Santaló type inequalities for the extremal general affine surface areas. This article may be regarded as an Orlicz extension of the recent work of Giladi, Huang, Schütt and Werner (2020), who introduced and studied the extremal Lp affine surface areas.

Some log-Minkowski inequalities for $$L_p$$-mixed affine surface area

Some log-Minkowski inequalities for $$L_p$$-mixed affine surface area

Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes

Consider a random simplex [X_1,ldots ,X_n] defined as the convex hull of independent identically distributed (i.i.d.) random points X_1,ldots ,X_n in mathbb {R}^{n-1} with the following beta density: Let J_{n,k}(beta ) be the expected internal angle of the simplex [X_1,ldots ,X_n] at its face [X_1,ldots ,X_k]. Define {tilde{J}}_{n,k}(beta ) analogously for i.i.d. random points distributed according to the beta' density {tilde{f}}_{n-1,beta } (x) propto (1+Vert xVert ^2)^{-beta }, xin mathbb {R}^{n-1}, beta > ({n-1})/{2}. We derive formulae for J_{n,k}(beta ) and {tilde{J}}_{n,k}(beta ) which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of beta . For J_{n,1}(pm 1/2) we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope K_{n,d} := [U_1,ldots ,U_n] where U_1,ldots ,U_n are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of K_{n,d}: lim _{nrightarrow infty } n^{-{({d-1})/({d+1})}}{mathbb {E}}{mathbf {f}}(K_{n,d}) = {mathbf {c}}_d cdot Omega (K), where Omega (K) is the affine surface area of K, and {mathbf {c}}_d is an unknown vector not depending on K. We compute {mathbf {c}}_d explicitly in dimensions up to d=10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.

Constrained convex bodies with extremal affine surface areas

Given a convex body K⊆Rn and p∈R, we introduce and study the extremal inner and outer affine surface areasISp(K)=supK′⊆K⁡(asp(K′)) and osp(K)=infK′⊇K⁡(asp(K′)), where asp(K′) denotes the Lp-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 IS1(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 Löwner ellipsoid to give asymptotic estimates on the size of ISp(K) and osp(K). Surprisingly, it turns out that both quantities are proportional to a power of volume.

Some inequalities for the (p, q)-mixed affine surface areas

The notion of (p, q)-mixed volume was first introduced by Lutwak, Yang and Zhang in 2018. According to this concept, Li, Wang and Zhou defined the (p, q)- mixed affine surface areas in 2019. In this paper, we establish some inequalities including cyclic inequality, monotonous inequality and product inequality for the (p, q)-mixed affine surface areas.

Difference inequalities for the i-th mixed affine surface areas

Difference inequalities for the i-th mixed affine surface areas

An $$\mathrm {SL}(n)$$-invariant mixed integral $$L_p$$ affine surface area

Extremal problems optimizing mixed integral $$L_p$$ affine surface area under linear transformation are investigated. The famous Lutwak–Yang–Zhang ellipsoid turns out to be a special case of the solutions. Associated John’s inclusions are also formulated for these new $$L_p$$ version John ellipsoids, which provide in turn new extensions of the classic John’s inclusion for John–Lowner ellipsoid.

A Steiner formula in the Lp Brunn Minkowski theory

We prove an analogue of the classical Steiner formula for the Lp affine surface area of a Minkowski outer parallel body for any real parameter 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 as well as the dual quermassintegrals of the dual Brunn Minkowski theory 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 Lp affine surface area of a Minkowski outer parallel body for any real parameters p and s is also given.

General Blaschke Bodies and the Asymmetric Negative Solutions of Shephard Problem

In this article, based on the Blaschke combination of convex bodies, we define the general Blaschke bodies and obtain the extremal values of their volume and affine surface area. Further, we study the asymmetric negative solutions of the Shephard problem for the projection bodies.

Lp Radial Blaschke-Minkowski Homomorphisms and Lp Dual Affine Surface Areas

Schuster introduced the notion of radial Blaschke-Minkowski homomorphism and considered the Busemann-Petty problem for volume forms. Whereafter, Wang, Liu and He presented the L p radial Blaschke-Minkowski homomorphisms and extended Schuster’s results. In this paper, associated with L p dual affine surface areas, we give an affirmative and a negative form of the Busemann-Petty problem and establish two Brunn-Minkowski inequalities for the L p radial Blaschke-Minkowski homomorphisms.

(p,q)-Mixed geominimal surface area and (p,q)-mixed affine surface area

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 Lp geominimal surface area and the Lp 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.

New affine inequalities and projection mean ellipsoids

A variational formula for the Lutwak affine surface areas $$\Lambda _{j}$$ of convex bodies in $$\mathbb {R}^n$$ is established when $$1\le j\le n-1.$$ By using introduced new ellipsoids associated with projection functions of convex bodies, we prove a sharp isoperimetric inequality for $$\Lambda _{j}$$ , which opens up a new passage to attack the longstanding Lutwak conjecture in convex geometry.

Halfspace depth and floating body

Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Maximum halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the maximum 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 of measures 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.

$L_p$-dual affine surface areas for the general L_p-centroid bodies

$L_p$-dual affine surface areas for the general L_p-centroid bodies

Lp-dual affine surface areas for the general Lp-intersection bodies

For 0 < p < 1, the notions of symmetric and asymmetric Lp-intersection bodies were introduced by Haberl and Ludwig. Recently, Wang and Li defined the general Lp-intersection bodies. In this paper, associated with the Lp-dual affine surface areas, we give the extremum values of the general Lp-intersection bodies. Moreover, a Brunn-Minkowski type inequality and a monotone inequality for the Lp-dual affine surface area version of general Lp-intersection bodies are established, respectively.