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?