  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.
