TITLE: Uniqueness for antipodal points of maximum distance on closed convex surface
QUESTION [6 upvotes]: Let $S\subset \mathbb{R}^3$ be a closed convex surface and let $p,q\in S$ be points such that $d(p,q)=\operatorname{diam}(S)$ where $\operatorname{diam}(S)$ is the diameter of $S$ with respect to the intrinsic distance $d$ of $S$. 
Does $d(p,q')=\operatorname{diam}(S)$ imply $q=q'$?

REPLY [4 votes]: Take two copies of an equilateral triangle $ABC$ and glue them along their boundary.  The resulting surface is homeomorphic to the sphere.  The metric is convex.  Then both $B$ and $C$ are at maximal distance from $A$ so $B$ is not the unique point at maximal distance from $A$.