Ákos G. Horváth

On Bisectors in Minkowski Normed Spaces

We discuss the concept of the bisector of a segment in a Minkowski normed n-space, and prove that if the unit ball K of the space is strictly convex then all bisectors are topological images of a hyperplane of the embedding Euclidean n-space. The converse statement is not true. We give an example in the three-space showing that all bisectors are topological planes, however K contains segments on its boundary. Strict convexity ensures the normality of Dirichlet-Voronoi-type K-subdivision of any point lattice.

Bounded representation and radial projections of bisectors in normed spaces

It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion combines the concepts of bisector and shadow boundary of the unit ball, both corresponding with the same spatial direction. The bounded representation visualizes the connection between the topology of bisectors and shadow boundaries (Lemma 1) and gives the possibility to simplify and to extend some known results on radial projections of bisectors. Our main result (Theorem 1) says that in the manifold case the topology of the closed bisector and the topology of its bounded representation are the same; they are closed,

Normally distributed probability measure on the metric space of norms

(n-1)

On the volume of the convex hull of two convex bodies

-dimensional balls embedded in Euclidean

Semi-Inner Products and the Concept of Semi-Polarity

n

Maximum volume polytopes inscribed in the unit sphere

-space in the standard way.

On the Dirichlet—Voronoi cell of unimodular lattices

Abstract In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of a measure with the property that its pushforward by the thinness function is a probability measure of truncated normal distribution. Finally, we improve this method to find a measure satisfying some important properties in geometric measure theory.

Extremal Polygons with Minimal Perimeter

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

On the sublattices of the Barnes-wall lattice

n