Let $K$ be a convex body in $\mathbb{R}^{n}$ . A diametral Santaló point of $K$ is defined as a minimizer of $\mathfrak{f}_{K}(u):={\mathrm{diam}} ((K-u)^{\circ })$ with respect to $u \in {\mathrm{int}}(K)$ . The set $\Phi (K)$ of diametral Santaló points of $K$ can be viewed as a sort of central region of $K$ . The study of this concept of centrality was initiated in a recent work of ours, entitled “Diametral Santaló points of convex bodies” and published in Aequationes Mathematicae. The present work analyzes the function $\mathfrak{f}_{K}$ and set $\Phi (K)$ under the assumption that $K$ is a polyhedral convex body. Polyhedrality is a key structural assumption in this paper. Exploiting this property enables a substantial extension of the theory beyond the general non-polyhedral case.
