Convex Bodies Research Articles

Groups of convex bodies

In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen’s polytope algebra, with the universal property that continuous valuations on convex bodies correspond to continuous homomorphisms on the group of convex bodies. To study this group, we first obtain a version of McMullen polynomiality for valuations that take values not in fields or vector spaces, but in abelian groups. Using this, we are able to equip the group of convex bodies with a grading that consists of real vector spaces in all positive degrees, mirroring one of the main structural properties of the polytope algebra. It is hoped that this work can serve as the starting point for a K-theoretic interpretation of valuations on convex bodies.

Stability related to the Lp$L_p$ Busemann–Petty problem

AbstractThe Busemann–Petty problem for asks: does imply for origin‐symmetric convex bodies K and L? Here, is the volume of K and the operator is the intersection body of K. In this paper, we study the linear stability and separation results related to the Busemann–Petty problem. A typical result we obtained is: for small enough, if K is an intersection body and L is a star body such that for any , the following inequality: implies where is the Minkowski functional of , and C is a positive number depending only on p and n. Moreover, we also prove the linear stability and separation results for the negative answer of the Busemann–Petty problem.

Truncated Log-concave Sampling for Convex Bodies with Reflective Hamiltonian Monte Carlo

We introduce Reflective Hamiltonian Monte Carlo (ReHMC), an HMC-based algorithm to sample from a log-concave distribution restricted to a convex body. The random walk is based on incorporating reflections to the Hamiltonian dynamics such that the support of the target density is the convex body. We develop an efficient open source implementation of ReHMC and perform an experimental study on various high-dimensional datasets. The experiments suggest that ReHMC outperforms Hit-and-Run and Coordinate-Hit-and-Run regarding the time it needs to produce an independent sample, introducing practical truncated sampling in thousands of dimensions.

Weighted Brunn-Minkowski Theory I: On weighted surface area measures

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in Rn – in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.

Nonlocal energies of convex bodies and their log-Minkowski problems

Under a pioneer work in integral geometry, Lutwak, Xi, Yang, and Zhang [19] established recently the variational formula for chord integral Iq(K), for q>0, and defined the q-th chord measure. Meanwhile, they provided sufficient and necessary conditions for the existence of a solution to the chord Minkowski problem, for q>0, and sufficient conditions to solve the symmetric case of the chord log-Minkowski problem when 1≤q≤n+1. It is well known that for q>1, Iq(K) is the Riesz potential of characteristic function of convex body K. In the case of 0<q<1, we discover the new relationship between nonlocal energy and chord integral Iq(K), which gives the representation formula of Iq(K) in the sense of Riesz potential, and we also give a new definition of chord measure by making use of its representation formula of Riesz potentials for 0<q<1. Finally, we solve the symmetric case of chord log-Minkowski problem under a sufficient condition for 0<q<1.

Convex Bodies With All Characteristics Planar

Abstract We show that smooth and strongly convex bodies in the symplectic $\mathbb R^{2n}$ for $n&amp;gt;1$ with all characteristics planar, or all outer billiard trajectories planar, are affine symplectic images of balls.

Extremizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes

In [20], I. Soprunov and A. Zvavitch have translated the Bezout inequalities (from Algebraic Geometry) into inequalities of mixed volumes satisfied by the simplex. They conjecture this set of inequalities characterizes the simplex, among all convex bodies in Rn. Together with C. Saroglou, they proved the characterization among all polytopes [17] and, for a larger set of inequalities, among all convex bodies [18]. The conjecture remains open for n≥4. In this work, we investigate necessary conditions on the structure of the boundary of a convex body K, for K to satisfy all inequalities. In particular, we obtain a new solution of the 3-dimensional case.

Mean surface and volume particle tensors under L-restricted isotropy and associated ellipsoids

Abstract The convex-geometric Minkowski tensors contain information about shape and orientation of the underlying convex body. They therefore yield valuable summary statistics for stationary marked point processes with marks in the family of convex bodies, or, slightly more specialised, for stationary particle processes. We show here that if the distribution of the typical particle is invariant under rotations about a fixed k-plane, then the average volume tensors of the typical particle can be derived from k + 1-dimensional sections. This finding extends the well-known three-dimensional special case to higher dimensions. A corresponding result for the surface tensors is also proven. In the last part of the paper we show how Minkowski tensors can be used to define three ellipsoidal set-valued summary statistics, discuss their estimation and illustrate their construction and use in a simulation example. Two of these, the so-called Miles ellipsoid and the inertia ellipsoid, are based on mean volume tensors of ranks up to 2. The third, based on the mean surface tensor of rank 2, will be called the Blaschke ellipsoid and is only defined when the typical particle has a rotationally symmetric distribution about an axis, as we then can use uniqueness and reconstruction results for centred ellipsoids of revolution from their rank-2 surface tensor. The latter are also established here.

Banach’s isometric subspace problem in dimension four

We prove that if all intersections of a convex body $B\subset \mathbb{R}^{4}$ with 3-dimensional linear subspaces are linearly equivalent then $B$ is a centered ellipsoid. This gives an affirmative answer to the case $n=3$ of the following question by Banach from 1932: Is a normed vector space $V$ whose $n$ -dimensional linear subspaces are all isometric, for a fixed $2 \le n< \dim V$ , necessarily Euclidean? The dimensions $n=3$ and $\dim V=4$ is the first case where the question was unresolved. Since the 3-sphere is parallelizable, known global topological methods do not help in this case. Our proof employs a differential geometric approach.

Minkowski's successive minima in convex and discrete geometry

In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel's lemma.

First Fossil of Tylidae (Isopoda: Oniscidea) in Kachin Amber, Myanmar, with a List of All Oniscidea Fossil Records

A fossil of Oniscidea, Tylidae gen. et sp. indet. from Kachin amber (Cretaceous Cenomanian), Myanmar, is described here. The convex body, the cephalon with a triangular protrusion between the antennae, and pereonites 2–6 with epimera demarcated from tergites indicate that this specimen belongs to the family Tylidae, but since it is not an adult the identification of the genus and species is uncertain. This specimen has a convex body and shows an ability to conglobate, like all Tylidae. It is the first specimen of Oniscidea with a conglobation ability found in Burmese amber. Up to now, the fossil record of terrestrial isopods has included a total of 20 families and 54 records (36 species and 18 not formally identified species), 20% of which are from the Cretaceous period. These fossil records from the Cretaceous period show that terrestrial isopods were highly diversified as early as in the Cenomanian.

Covering a reduced spherical polygon by a disk

A spherical convex body R is called reduced if any convex body properly contained in R has thickness smaller than the thickness Δ(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (R)$$\\end{document} of R. We show that for every reduced spherical convex polygon V there exists its boundary point such that the disk of radius Δ(V)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (V)$$\\end{document} centered at this point contains V.

A Practical Algorithm for Volume Estimation based on Billiard Trajectories and Simulated Annealing

We tackle the problem of efficiently approximating the volume of convex polytopes, when these are given in three different representations: H-polytopes, which have been studied extensively, V-polytopes, and zonotopes (Z-polytopes). We design a novel practical Multiphase Monte Carlo algorithm that leverages random walks based on billiard trajectories, as well as a new empirical convergence test and a simulated annealing schedule of adaptive convex bodies. After tuning several parameters of our proposed method, we present a detailed experimental evaluation of our tuned algorithm using a rich dataset containing Birkhoff polytopes and polytopes from structural biology. Our open-source implementation tackles problems that have been intractable so far, offering the first software to scale up in thousands of dimensions for H-polytopes and in the hundreds for V- and Z-polytopes on moderate hardware. Last, we illustrate our software in evaluating Z-polytope approximations.

Concurrent normals of immersed manifolds

It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be true for $n=2,3,4$. Motivated by a recent results of Y. Martinez-Maure, and an approach by A. Grebennikov and G. Panina, we prove the following: Let a compact smooth $m$-dimensional manifold $M^m$ be immersed in $ \mathbb{R}^n$. We assume that at least one of the homology groups $H_k(M^m,\mathbb{Z}_2)$ with $k&lt;m$ vanishes. Then under mild conditions, almost every normal line to $M^m$ contains an intersection point of at least $\beta +4$ normals from different points of $M^m$, where $\beta$ is the sum of Betti numbers of $M^m$.

Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p-center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented.

A Note on Stabbing Convex Bodies with Points, Lines, and Flats

A Note on Stabbing Convex Bodies with Points, Lines, and Flats

Diversities and the Generalized Circumradius

The generalized circumradius of a set of points Asubseteq mathbb {R}^d with respect to a convex body K equals the minimum value of lambda ge 0 such that a translate of lambda K contains A. Each choice of K gives a different function on the set of bounded subsets of mathbb {R}^d; we characterize which functions can arise in this way. Our characterization draws on the theory of diversities, a recently introduced generalization of metrics from functions on pairs to functions on finite subsets. We additionally investigate functions which arise by restricting the generalized circumradius to a finite subset of mathbb {R}^d. We obtain elegant characterizations in the case that K is a simplex or parallelotope.

A Characterization of the Polarity Mapping for Convex Bodies

In this paper, we establish a characterization of the polarity mapping for 1-dimensional convex bodies, which is a supplement to the result for such a characterization obtained by Böröczky and Schneider.

Investigation of Convex Bodies in $${\mathbf{R}}^{\mathbf{3}}$$ by Support Planes

Investigation of Convex Bodies in $${\mathbf{R}}^{\mathbf{3}}$$ by Support Planes

Spherical centroid bodies

The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions---one geometric, the other probabilistic in nature---are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann--Petty centroid inequality.