Ellipsoid characterization theorems

Abstract

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 int K 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.

The equal tangents property

“…This statement is easily proved by elementary geometry. The result was extended to the case of Minkowski planes by S. Wu [9], and Z. Lángi [3] also gave a characterization of the ellipsoid among centrally symmetric convex bodies in terms of tangent segments of equal Minkowski length.…”