Related papers: Convex bodies generated by sublinear expectations …
We construct triangular hyperbolic polyhedra whose links are generalized 4-gons. The universal cover of those polyhedra are hyperbolic buildings, which appartments are hyperbolic planes tesselated by regular triangles with angles $\pi/4$.…
Conformal transformations of a Euclidean (complex) plane have some kind of completeness (sufficiency) for the solution of many mathematical and physical-mathematical problems formulated on this plane. There is no such completeness in the…
We study isomorphic properties of two generalizations of intersection bodies, the class of k-intersection bodies and the class of generalized k-intersection bodies. We also show that the Banach-Mazur distance of the k-intersection body of a…
Makeev conjectured that every constant-width body is inscribed in the dual difference body of a regular simplex. We prove that homologically, there are an odd number of such circumscribing bodies in dimension 3, and therefore geometrically…
Any solid object can be decomposed into a collection of convex polytopes (in short, convexes). When a small number of convexes are used, such a decomposition can be thought of as a piece-wise approximation of the geometry. This…
We discuss a construction that gives counterexamples to various questions of unique determination of convex bodies.
This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.
Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a…
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.
Many star bodies have convex subsets with approximately the same Gaussian measure (of the complement). Inspired by this phenomenon, and in connection with the randomized Dvoretzky theorem for Lorentz spaces, we derive bounds on the…
{We show in this paper that two normal elliptic sections through every point of the boundary of a smooth convex body essentially characterize an ellipsoid and furthermore, that four different pairwise non-tangent elliptic sections through…
Recently, Bo'az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag's quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies…
We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at…
Elasticity ellipses or central ellipses have been long used in graphic statics to capture the elastic behaviour of structural elements. The paper gives a generalisation the concept both in dimensions and in the possibility of degenerate…
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex…
It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra…
This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…
The parameterisation of rotations in three dimensional Euclidean space is an area of applied mathematics that has long been studied, dating back to the original works of Euler in the 18th century. As such, many ways of parameterising a…
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This…
The average mixed volume of a random projection of $d$ convex bodies in $\mathbb R^n$ is bounded above in terms of a quermassintegral.