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

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