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.
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