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The contact polytopes of the Barnes-Wall lattices in 1, 2, 4, and 8 dimensions are all uniform polytopes. Is this true in any higher number of dimensions?

Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$. Question: What are the edges of $P$? Let'...