Zsolt Lángi

Ball-Polyhedra

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.

On the volume of the convex hull of two convex bodies

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean

On diagonalizable operators in Minkowski spaces with the Lipschitz property

n

Maximum volume polytopes inscribed in the unit sphere

n-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.

Ball and spindle convexity with respect to a convex body

Our goal is to identify the type and number of static equilibrium points of solids arising from fine, equidistant n-discretizations of smooth, convex surfaces. We assume uniform gravity and a frictionless, horizontal, planar support. We show that as n approaches infinity these numbers fluctuate around specific values which we call the imaginary equilibrium indices associated with the approximated smooth surface. We derive simple formulae for these numbers in terms of the principal curvatures and the radial distances of the equilibrium points of the solid from its center of gravity. Our results are illustrated on a discretized ellipsoid and match well the observations on natural pebble surfaces.

ON THE BEZDEK-PACH CONJECTURE FOR CENTRALLY SYMMETRIC CONVEX BODIES

Abstract semi-inner-product space is a real vector space M equipped with a function [ . , . ] : M × M → R which is linear in its first variable, strictly positive and satisfies the Schwartz inequality. It is well known that the function | | x | | = [ x , x ] defines a norm on M and vice versa, for every norm on X there is a semi-inner-product satisfying this equality. A linear operator A on M is called adjoint abelian with respect to [ . , . ] , if it satisfies [ Ax , y ] = [ x , Ay ] for every x , y ∈ M . The aim of this paper is to characterize the diagonalizable adjoint abelian operators in finite dimensional real semi-inner-product spaces satisfying a certain smoothness condition.

Density bounds for outer parallel domains of unit ball packings

The lack of an inner product structure in Banach spaces yields the motivation to introduce a semi-inner product with a more general axiom system, one missing the requirement for symmetry, unlike the one determining a Hilbert space. We use it on a finite dimensional real Banach space