Parker Glynn-Adey

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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