Ellipsoid characterization theorems

TL;DR

This paper proves two theorems characterizing ellipsoids in normed spaces, showing conditions under which convex bodies must be ellipsoids based on tangent segment properties and billiard angular bisectors.

Contribution

It introduces new ellipsoid characterization theorems in normed spaces using tangent segment and billiard bisector conditions.

Findings

Convex body with tangent segment property is a homothetic ellipsoid.

Strictly convex smooth norm unit ball with billiard bisectors as Busemann or Glogovskij bisectors is an ellipse.

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 $p \notin K$ and in any 2-dimensional plane $P$ intersecting $\inter K$ 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.