Antipodal points after Vîlcu

I’ve been thinking a lot about convex bodies in 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 be a centrally symmetric convex body in . Let denote thes intrinsic metric of and its intrinsic diameter. For a point we write for its image under the central symmetry.

Lemma 1 (Vîlcu) If then .

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.