Let $S$ be the surface of a convex body, polyhedral or smooth, embedded in $\mathbb{R}^3$. For a point $x \in S$, let $F(x)$ be the set of furthest points from $x$, measured by shortest paths on the surface $S$. Let $f(x)$ be the length of those shortest paths: $|x y|$ for $y \in F(x)$.

It seems natural to hope that

Hypothesis: For any $x \in S$, $f(x) \ge \tfrac{1}{2} \mathrm{diam}(S)$.

Here $\mathrm{diam}(S)$ is the maximum distance between any two points on $S$ (again measured by shortest paths on the surface of $S$). Suppose, for example, that $\rho$ is a diameter-realizing geodesic. Then for any $x \in \rho$, $f(x) \ge \tfrac{1}{2} |\rho|$, just tracking along $\rho$.

A non-comprehensive literature search has failed to uncover a relationship between $f(x)$ and $\mathrm{diam}(S)$.

Itoh, Jin‐ichi and Costin Vǐlcu. "Criteria for farthest points on convex surfaces."

*Mathematische Nachrichten*282, no. 11 (2009): 1537-1547. Journal link.