Averkov, Gennadiy : A monotonicity lemma for bodies of constant Minkowskian width

Author(s):

Averkov, Gennadiy

Title:

A monotonicity lemma for bodies of constant Minkowskian width

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 16, 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:

A finite dimensional normed linear space is usually called a Minkowski space. Suppose that for the parameter $t$ changing from 0 to 1 two points $p_1(t)$ and $p_2(t)$ traverse the boundary of a convex body $K$ in a Minkowski plane with respect to counterclockwise and clockwise orientation, respectively, and that they do not coincide for each $t.$ We prove that $K$ is of constant Minkowskian width if and only if the Minkowskian distance between $p_1(t)$ and $p_2(t)$ is a unimodal function of $t$ for each possible choice of the functions $p_1(t)$ and $p_2(t).$ Furthermore, we give an appropriate extension of the ``only if'' part of this result to higher dimensional Minkowski spaces.

Keywords:

body of constant (Minkowskian) width, Minkowski space, monotonicity lemma

Language:

English

Publication time:

12 / 2002