# The set of non-smooth points of a convex function is (m - 1)-rectifiable

I am looking for a reference to the following result.

Let $f:\mathbb R^m\to\mathbb R$ be a convex function. Then $f$ is differentiable at all points of outside of a countable union of $(m-1)$-rectifiable sets.

Comments:

• $n$-rectifiable set is an image of Lipschitz map from bounded domain in $\mathbb R^n$

• I checked Federer's "Geometric Measure Theory", but I might miss the right place.

• Extract from the Greg's answer (for those who are lazy to read the paper): In this paper, it is given a complete characterization of subsets of nondifferatiable points of a convex function. Namely, it is proved that $A$ is a such a set if and only if it can be covered by countably many graphs of DC-functions. ("DC-function" = "Difference of Convex functions".)

-

## 1 Answer

The paper On the differentiation of convex functions in finite and infinite dimensional spaces by Zajíček primarily deals with the general question in Banach space, but it looks like it has a summary of the situation (as of 1979) that tells you everything that you might want to know. It has references to closer papers by Anderson-Klee and Besicovitch.

I just Googled around and eventually got to this paper.

-
Thank you so much :) – Anton Petrunin May 23 '10 at 22:09