The mean width of a convex body is the average distance between parallel supporting hyperplanes when the normal direction is chosen uniformly over the sphere. The Simplex Mean Width Conjecture (SMWC) is a longstanding open problem that says the regular simplex has maximum mean width of all simplices contained in the unit ball and is unique up to isometry. We give a self contained proof of the SMWC in $d$ dimensions. The main idea is that when discussing mean width, $d+1$ vertices $v_i\in\mathbb{S}^{d-1}$ naturally divide $\mathbb{S}^{d-1}$ into $d+1$ Voronoi cells and conversely any partition of $\mathbb{S}^{d-1}$ points to selecting the centroids of regions as vertices. We will show that these two conditions are enough to ensure that a simplex with maximum mean width is a regular simplex.

翻译：凸体的平均宽度是指当法向量在球面上均匀选取时，平行支撑超平面之间的平均距离。单纯形平均宽度猜想（SMWC）是一个长期未解决的开放问题，它指出所有包含于单位球内的单纯形中，正则单纯形具有最大的平均宽度，且在等距意义下唯一。我们给出了一个关于 SMWC 在 $d$ 维空间中的自包含证明。主要思想是：在讨论平均宽度时，$d+1$ 个顶点 $v_i\in\mathbb{S}^{d-1}$ 自然地将 $\mathbb{S}^{d-1}$ 划分为 $d+1$ 个沃罗诺伊细胞。反过来，$\mathbb{S}^{d-1}$ 的任意划分也暗示了选择这些区域的质心作为顶点。我们将证明这两个条件足以确保具有最大平均宽度的单纯形是正则单纯形。