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：convex 体的平均值宽度是,当正常方向在球体上统一选择时,平行支持超高空平面的平均距离。 SMWC 是一个长期的开放问题, 表明正单面光线具有单位球内所有implices的最大平均宽度, 且在异度上是独一无二的。 我们用美元维度来提供 SMWC 的自封证明。 主要的想法是, 当讨论平均宽度时, $d+1$ vertics $v_ i\ in\ mathb{S ⁇ d-1} $ 自然将 $\mathbb{S ⁇ d-1} 等于 $d+1$ Vorononoi 单元格和 $d+1$@sd-1} 等分差值, 反之, 选择区域偏差为垂直点。 我们将会显示这两个条件足以确保最平均宽度的简单x是一个常规简单值。