Reference for the rectifiablity of the boundary hypersurface of convex open set

Question

The boundary of any convex open set $X$ is $\mathbb R^n$ is a rectifiable hypersurface.

To see this, intuitively, simply take a sphere $S_d$ with diameter $d\in(0,+\infty]$ that contains $X$. The nearest point projection from $S_d$ to $\partial X$ is one-to-one onto.

Although the rectifiability result is not hard and well-known, I am having a hard time finding the reference to cite. Could you please help me with it? Just any reference/textbook would be fine and I will go from there.

When writing a research paper and stating a result, I think I need to try the best to find the earliest possible reference.

Answer 1

Any convex function is Lipschitz continuous so the boundary of a convex set is locally a graph of a Lipschitz function and therefore it is rectifiable.

For a proof of Lipschitz continuity of convex functions, see for example Theorem 2.31 in:

B. Dacorogna, Direct methods in the calculus of variations. Second edition.