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

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
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