Averkov, Gennadiy : Constant Minkowskian width in terms of double normals

Author(s):

Averkov, Gennadiy

Title:

Constant Minkowskian width in terms of double normals

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 15, 2002

Mathematics Subject Classification:

52A21			[ Finite-dimensional Banach spaces ]
52A38			[ Length, area, volume ]
52A10			[ Convex sets in $2$ dimensions ]
52A20			[ Convex sets in $n$ dimensions ]

Abstract:

We extend the notion of a double normal of a convex body from smooth strictly convex Minkowski spaces to arbitrary real, normed, linear spaces in two different ways. Then for both of the ways we obtain the following characterization theorem: a convex body $K$ in a Minkowski plane is of constant Minkowskian width iff every chord $I$ of $K$ splits it into two compact convex sets $K_1$ and $K_2,$ such that $I$ is a Minkowskian double normal of $K_1$ or $K_2.$ Furthermore, this theorem applied to the Euclidean plane is extended to $d$-dimensional Euclidean spaces.

Keywords:

body of constant (Minkowskian) width, Minkowski space, double normal, section, hyperplane

Language:

English

Publication time:

12 / 2002