A characterization of the ellipsoid through planar grazes

TL;DR

This paper characterizes ellipsoids among convex bodies in three-dimensional space by examining planar grazes from boundary points, showing that under certain symmetry and convexity conditions, the inner body must be an ellipsoid.

Contribution

It provides a new geometric characterization of ellipsoids based on properties of planar grazes and convexity conditions, extending previous understanding.

Findings

If two symmetric convex bodies satisfy the grazing condition, the inner one is an ellipsoid.

The result relies on the bodies being strictly convex and almost free with respect to each other.

The characterization applies to bodies in three-dimensional space with specific symmetry and convexity assumptions.

Abstract

In this paper we proved the following: \emph{Let $K, L\subset \mathbb R^3$ be two $O$-symmetric convex bodies with $L\subset \emph{int} K$ strictly convex. Suppose that from every $x$ in $\emph{bd} K$ the graze $\Sigma(L,x)$ is a planar curve and $K$ is almost free with respect to $L$. Then $L$ is an ellipsoid.}