Averkov, Gennadiy : On planar convex bodies of given Minkowskian thickness and least possible area

Author(s):

Averkov, Gennadiy

Title:

On planar convex bodies of given Minkowskian thickness and least possible area

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 11, 2003

Mathematics Subject Classification:

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

Abstract:

Let $K$ be a convex body in a Minkowski plane, i.e., in a two-dimensional real Banach space. The Minkowskian thickness of $K$ is the minimal possible Minkowskian distance between two points of $K$ lying in different parallel supporting lines of that convex body. Let $X$ be the class of planar convex bodies having a given Minkowskian thickness, say one, and least possible area. We prove that each body $K$ from $X$ is necessarily a triangle or a quadrilateral. Furthermore, under certain conditions involving the Minkowskian unit ball, the class $X$ consists only of triangles. The result of P\'al \cite[\S10]{MR49:9736}, stating that in Euclidean case $X$ is the class of equilateral triangles with altitudes of length one, is obtained as a simple consequence of our main theorem.

Keywords:

Banach space, normed space, Minkowski space, geometric inequality, Blaschke diagram, thickness, cross-section measure

Language:

English

Publication time:

12 / 2003