Ellipsoid characterization theorems

Lángi, Zsolt (2013) Ellipsoid characterization theorems. Advances in Geometry, 13 (1). pp. 145-154. ISSN 1615-7168

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Abstract

In this note we prove two ellipsoid characterization theorems. The first one is that if K is a convex body in a normed space with unit ball M, and for any point p ∉ K and in any 2-dimensional plane P intersecting intK and containing p, there are two tangent segments of the same normed length from p to K, then K and M are homothetic ellipsoids. Furthermore, we show that if M is the unit ball of a strictly convex, smooth norm, and in this norm billiard angular bisectors coincide with Busemann angular bisectors or Glogovskij angular bisectors, then M is an ellipse.

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