Sabok的S-单纯形问题的有限与Urysohn阻碍
Finite and Urysohn obstructions to Sabok's S-prime simplex questions
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中文总结 AI 辅助
针对Sabok提出的S'-单纯形相关的两个问题，本文证明直径不超过1的可分度量空间对应的S'(X)不一定是单纯形，S'(U₁)也不是Poulsen单纯形，还给出了有限空间和Urysohn球情形下的具体阻碍与判定依据。
中文摘要 AI 辅助
Sabok提出了两个问题：附于直径不超过1的可分度量空间的紧凸集\boldsymbol{S'(X)}是否总是单纯形，以及\boldsymbol{S'(\boldsymbol{\bigcup}_1)}是否为Poulsen单纯形。本文给出否定回答。对于有限空间\boldsymbol{X=\boldsymbol{\bigcup}_1,\boldsymbol{\bigcup}_m}，\boldsymbol{S'(X)}与距离矩阵行\boldsymbol{r_i=(d(x_i,x_1),\boldsymbol{d(x_i,x_m)})}的凸包仿射同胚，当且仅当这些行仿射独立时它是单纯形，直径为1的4-循环给出最小有限阻碍。对于Urysohn球，以有理Urysohn球\boldsymbol{D}为坐标，本文将坐标模型\boldsymbol{S'_D(\boldsymbol{\bigcup}_1)}识别为Katětov紧空间\boldsymbol{K(D)}，四个显式极点\boldsymbol{f_A,g_A,\boldsymbol{1},\boldsymbol{h}}满足\boldsymbol{f_A+g_A=\boldsymbol{1}+\boldsymbol{h}}，给出\boldsymbol{(3/4)\boldsymbol{1}}的两个不同表示测度，故\boldsymbol{S'(\boldsymbol{\bigcup}_1)}不是Choquet单纯形。
英文摘要
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.
