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

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 $$din(0,+infty]$$ 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.

MathOverflow Asked on December 28, 2020

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.

Correct answer by Piotr Hajlasz on December 28, 2020

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