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Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming

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two tetrahedra in $\mathbb R^4$

It is relatively easy to show (see below) that if we have two equilateral triangles of side 1 in $\mathbb R^3$, such that their union has diameter $1,$ then they must share a vertex. I wonder whether …
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