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A convex $d$-polytope $P$ is the convex hull of finitely many points. Given such a polytope with $n \gg d$ vertices, I would like to prove that its surface has to be "round" in some region.

Let me try to explain what my intuition behind this speculation is, so that it maybe also becomes clearer what I mean by "round": If $P$ has a lot of vertices, there must be a point on the surface that has many vertices in a small neighbourhood of it. Many vertices in a small neighbourhood should mean that the polytope looks in this region almost like a smooth convex body. Of course this is not only incorrect since the polytope is still discrete but also because one could have something like a bipyramid over an $n$-gon, with $n$ very large again. In this case one only had an arc on the surface that is almost smooth but not a $(d-1)$-dimensional neighborhood.

I am not a mathematician by training. This is why I have no idea how to even measure "roundness" of such a region. But I suspect that mathematicans have precise notions that express formally what I only can explain intuitively.

Any hints for what I am actually looking for? Where should I start reading?

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I'm trying to understand... It's certainly true that the surface of a cube contains perfectly round circles: I can draw a bunch of circles on the surface. Presumably this is not what you meant. Maybe you meant that some faces of the polytope will be almost round, like a nearly regular 100-gon? –  user31373 Jun 7 '12 at 15:56
    
@LeonidKovalev: I edited the question and added a few explanations for clarification. –  Gregor Jun 7 '12 at 20:40
    
+1, nice question! This might be a good way to start: The Gauss map associates each face of the polytope with a point on the $(d-1)$-sphere, and each vertex with a convex region on it. Thus, you are dividing the area of the sphere into $n$ parts; when $n$ is large, some of these regions will have to be small. Of course, they could be long and skinny, as in the case of a flat $n$-gon. –  Rahul Jun 7 '12 at 20:56
    
It seems that the question is closest to the area of Random Polytopes, where one chooses vertices according to some probability distribution and asks what properties the convex hull will have with high probability. For example, if the vertices are chosen with uniform probability from a convex smooth body, the polytope will likely be "close" to this convex body. Two sample articles: arxiv.org/abs/0906.0309, arxiv.org/abs/math/0610192 .. But I don't know any result concerning approximate smoothness of the kind you described. You could ask Imre Barany, renyi.hu/~barany –  user31373 Jun 7 '12 at 20:57
    
there is the banach-mazur distance to try to quantify what being close to means in this context. however, it's a global distance and your problem is local. also, it's a bit weird : if you try to formalize it, you want to get close to something that is smooth, but you have to exclude a smooth face or else it's trivial. I think it might be better to consider this problem first : consider a sequence of polytopes with more and more vertices everywhere. it should get close to a smooth convex set. –  Glougloubarbaki Jun 7 '12 at 21:43

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