I recently reported about a polyhedral proof of Stoker’s theorem using a lemma of Martin Winter; this, in turn, uses technique by Izmestiev.

I wanted to take a brief moment to illustrate that technique; Gromov, a few years ago, published a marvelous little paper. He used differential geometry techniques to show that a polytope cannot be too long in every direction. Specifically: Consider P a d-dimensional polytope, and map it cellularly to a d-cube C. Pick two opposing facets , of that d-cube. Their preimages in P are d-1-disks.

Nice enough. Now, try to go from the preimage of to the preimage of . If you go from one facet to an adjacent one in , then you “pay” the angle between their normals. All together, the optimal way you can take gives a minimal angular distance between the preimage of to the preimage of in ; it describes how far the two preimages are apart.

Now, it would be nice, and perhaps point towards the polynomial Hirsch conjecture, if we could say something about this distance. But Gromov says: geometry naturally gives something different: If I minimize over ALL the opposing pairs of facets of , then I obtain , the angular spread… it describes the question whether P is long in every direction. Gromov’s paper obtains a displayed bound by passing through smooth mean-curvature geometry [1]. He needs to smoothen the polytope first, and his argument for non-simple polytopes is a bit … fishy. In comes Izmestiev, and I want to illustrate his method here, and improve the bound using it.

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