Euclidean Realizability of Fundamental-Polytope Types

Abstract

For a finite metric d, its fundamental polytope is the convex hull of the vectors (e_i-e_j)/d_ij. We answer the universality clause of Vershik's question about Euclidean metrics: not every abstract combinatorial type is Euclidean. The four-cycle metric gives a cube, whereas four Euclidean points give a fundamental polytope with 6, 10, or 12 vertices, never 8. Four points is minimal. In contrast, every strict metric has a Euclidean representative of the same labelled type. Adding a common constant c to its off-diagonal distances preserves its face lattice, and c >= 2n diam(d) guarantees an affinely independent Euclidean realization. The argument combines the known four-cycle cube, the Gordon-Petrov face criterion, and the classical additive-constant embedding method. It does not classify all degenerate Euclidean types or close the full source record. This is an AI-assisted, self-audited, unrefereed preprint; no absolute priority claim, independent review or formal verification is asserted. Source problem: Vershik (2015), Problem 1(4), DOI 10.1007/s40598-014-0005-z; UnsolvedMath record AMR-040-0001.