Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere?

Question

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere?

I want to know if I have any convex subset $M$ of $\mathbb{R}^n$ is it meaningful to talk about integral of second kind over $\partial M$?

By integral of second kind I mean integral where you use normal of the surface.

Answer 1

The answer appears to be yes -- indeed, it is possible to define the second derivative almost everywhere, in the sense that the corresponding second-order Taylor expansion is an $o(h^2)$ approximation. The standard reference is:

A. D. Alexandrov, “Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it,” Leningrad Sate University Annals [Uchenye Zapiski], Mathematical Series, 6 (1939), 3–35 (in Russian).

Unfortunately, this does not seem to be available online. Another key reference is:

H. Busemann and W. Feller, Krümmungseigenschaften Konvexer Flächen. (German) Acta Math. 66 (1936), no. 1, 1–47.

This one is available online, but is written in German.

Based on descriptions of these papers in other sources, the proof is to first show that every convex function has such a second-order expansion almost everywhere, and then show that the boundary of any convex set coincides locally with the graph of a convex function.