## 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.

Proof: Let and suppose, for contradiction, that . Pick some length minimizing geodesic connecting to . Let denote the concatenation of the paths and .

We check that is self-intersection free. Certainly is self-intersection free, because it is a minimizing geodesic. Suppose that . Then we have two minimizing geodesics from to intesecting at a point on their interior. Hence they must coincide, a contradiction. Thus is self-intersection free and hence seperates into two open regions. Let . We know since the central symmetry has to swap the components.

Suppose, for contradiction, that . Then . If then , contradicting our hypothesis on . If then we contradict the maximality of the diameter. Without loss of generality, take . We have . Take a minimizing geodesic joining to . We have that by the Jordan curve theorem. Take . Suppose .

The triangle inequality gives us:

Thus:

The maximality of the diameter and the equality then give:

Suppose that and . We then contradict the equality above. Thus, one of the two strict inequalities is an equality. Suppose without loss of generality that . Let denote a minimizing geodesic segment connecting to .

We then have that , otherwise we’ll have two geodesics diverging from one another at a point. We then form by removing from and replacing it with . We then have that and diverge at a point, a contradiction.

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