Two applications of topology to convex geometry. (English) Zbl 1104.52001
Geometric topology and set theory. Collected papers. Dedicated to the 100th birthday of Professor Lyudmila Vsevolodovna Keldysh. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica. Proceedings of the Steklov Institute of Mathematics 247, 164-167 (2004) and Tr. Mat. Inst. Steklova 247, 182-185 (2004).
Summary: The purpose of this paper is to prove two theorems of convex geometry using the techniques of topology. The first theorem states that if, for a strictly convex body \(K\), one may choose continuously a centrally symmetric section, then \(K\) must be centrally symmetric. The second theorem states that if every section of a three-dimensional convex body \(K\) through the origin has an axis of symmetry, then there is a section of \(K\) through the origin which is a disk.
For the entire collection see [Zbl 1087.55002].
For the entire collection see [Zbl 1087.55002].
MSC:
| 52A15 | Convex sets in \(3\) dimensions (including convex surfaces) |
