Kupitz–Martini–Perles conjecture on standard ball polyhedra

Let $V\subset\mathbb{R}^3$ be an extremal set for the frequent large distance problem, and let $\mathcal{B}(V)$ denote the associated ball polyhedron. The set $V$ is strongly critical when it has no extremal configuration as a proper subset. Kupitz–Martini–Perles conjecture.

$$\mathcal{B}(V)\text{ is a standard ball polyhedron if and only if }V\text{ is strongly critical}.$$

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