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
