Finite and Urysohn obstructions to Sabok's S-prime simplex questions

Yutong Zhang; Yaoran Yang

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arXiv · 2607.27215

Finite and Urysohn obstructions to Sabok's S-prime simplex questions

Yutong Zhang·Yaoran Yang

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Abstract

Sabok asked whether the compact convex set $S'(X)$ attached to a separable metric space of diameter at most one is always a simplex, and whether $S'(\mathbb U_1)$ is the Poulsen simplex. We give negative answers. For finite $X=\{x_1,\ldots,x_m\}$, $S'(X)$ is affinely homeomorphic to the convex hull of the rows $r_i=(d(x_i,x_1),\ldots,d(x_i,x_m))$ of the distance matrix; it is a simplex exactly when these rows are affinely independent. The diameter-one four-cycle gives the minimal finite obstruction. For the Urysohn sphere, using the rational Urysohn sphere $D$ as coordinates, we identify the coordinate model $S'_D(\mathbb U_1)$ with the Katétov compactum $K(D)$. Four explicit extreme points $f_A,g_A,\mathbf 1,\mathbf h$ satisfy $f_A+g_A=\mathbf 1+\mathbf h$, giving two distinct representing measures for $(3/4)\mathbf 1$. Hence $S'(\mathbb U_1)$ is not a Choquet simplex.

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Yutong Zhang, Yaoran Yang. 2026-06-18. Finite and Urysohn obstructions to Sabok's S-prime simplex questions. https://arxiv.org/abs/2607.27215

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