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

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.