## Antipodal points after Vîlcu

Posted in Math by pgadey on 2013/03/26

I’ve been thinking a lot about convex bodies in ${{\mathbb R}^3}$ lately. This post is going to be a write up of a useful lemma in the paper: Vîlcu, Constin, On Two Conjectures of Steinhaus, Geom. Dedicata 79 (2000), 267-275.

Let ${S}$ be a centrally symmetric convex body in ${{\mathbb R}^3}$. Let ${d(x,y)}$ denote thes intrinsic metric of ${S}$ and ${D = \sup_{x,y} d(x,y)}$ its intrinsic diameter. For a point ${x \in S}$ we write ${\bar{x}}$ for its image under the central symmetry.

Lemma 1 (Vîlcu) If ${d(x,y) = D}$ then ${y = \bar{x}}$.

This lemma says that if a pair realizes the inner diameter of a centrally symmetric convex body, the pair has to be centrally symmetric. This aligns well with our intuition about the sphere and cube, for example.

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