Let $\{K_i\}$ be a sequence of convex compact $n$-dimensional subsets in a Euclidean space $\mathbb{R}^n$. Assume $\{K_i\}$ converges in the Hausdorff metric to a convex compact set $K$ which is also $n$-dimensional.

Consider the sequence of boundaries $\{\partial K_i\}$ equipped with the induced intrinsic (!) metrics.

Is it true that the sequence $\{\partial K_i\}$ with these intrinsic metrics converges to $\partial K$ (again, equipped with the intrinsic metric) in the Gromov-Hausdorff sense?

A proof or a reference will be helpful.


The reference is Lemma 10.2.7 in A course of metric geometry by Burago-Burago-Ivanov. They do it in 3d but it does not matter. The main point is that if two convex bodies are Hausdorff close, then one can blow up one of them by slight dilation to contain the other one. Then the nearest point projection lets you project paths on the blown up convex body onto the other one. This easily implies that the intrinsic metrics on the boundaries are close uniformly, and hence GH close.


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