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> The boundary of any convex open set $X$ is $\mathbb R^n$ is a rectifiable hypersurface.
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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](https://mathoverflow.net/questions/169237/boundary-of-an-open-bounded-and-convex-set-in-mathbbr-n), 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.



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


