Convex bodies with centrally symmetric sections

TL;DR

The paper studies Barker-Larman’s ellipsoid characterization in the setting of central symmetry. It defines a jointly sufficient framework with a central ball $B$ inside $K$ that is suitable for $K$ and requires that every supporting hyperplane of $B$ cuts $K$ into a centrally symmetric section; under these hypotheses, if $K$ is centrally symmetric and strictly convex with center $O$ and $O\notin B$, then $K$ must be an ellipsoid. The authors prove the result first in dimension $3$, showing that plentiful shadow boundaries and affine axes of symmetry force $K$ to be an ellipsoid, and then extend the argument to $n>3$ via a projection-induction method. This work advances the Barker-Larman program by confirming a significant special case where central symmetry yields ellipsoidality.

Abstract

Let $K\subset \mathbb{R}^n$ be a convex body, $n\geq 3$. We say that $K$ satisfies the Barker-Larman condition if there exists a ball $B$ in the interior of $K$ such that for every suppor hyperplane $Π$ of $B$, the section $Π\cap K$ is a centrally symmetric set. Barker and Larman conjectured that the Barker-Larman condition characterizes the ellipsoid. In this work we prove an special case of such conjecture, in particular, we assume that the convex body $K$ is centrally symmetric. Our main result is the following: Let $K$ be a centrally symmetric and strictly convex body, with center at $O$, and let $B$ be a ball in the interior of $K$ and not containing $O$: If $K$ satisfies the Barker-Larman condition with respect to $B$ and $B$ is suitable for $K$ (intuitively, $B$ is suitable for $K$ if the boundary of $B$ is not very close to the boundary of $K$), then $K$ is an ellipsoid.

Figures (7)

• Figure 1: The circle $\mathbb{S}^1$ has suitable circle.
• Figure 2: The chords $a_ib_i$ converge to a common chord of $B$ and $-B$.
• Figure 3: The shadow boundary $S\partial(K,u)$ is contained in a plane.
• Figure 4: The line $L_{\Sigma}$ is affine axis of symmetry of $K$.
• Figure 5: The curve $\delta_{v_0}=\gamma_{v_0}$ is the union of two simple closed con-ti-nuous curves $\gamma_{v_0}^1$, $\gamma_{v_0}^2$.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2: Olevjanischnikoff
• Theorem 3: Rogers
• Definition 1
• Remark 1
• Definition 2
• Claim 1
• Remark 2
• proof
• Lemma 1