# Does there exist a convex function which is not strictly differentiable almost everywhere?

It is known that:

(i) There exists a locally Lipschitz function on an Euclidean finite-dimensional space which is almost everywhere (almost everywhere means "except a set with Lebesgues measure zero") Frechet differentiable but is not almost everywhere strictly differentiable.

(ii) Any convex function on an Euclidean finite-dimensional space is strictly differentiable except a set of the first category.

(iii) There exists a set of the first category on $\mathbb R$, the complement of which has Lebesgues measure zero.

I have a question: Does there exist a convex function on an Euclidean finite-dimensional space which is not almost everywhere strictly differentiable?

Note that for a convex function, strictly differentiable at a point means that its subdifferential of convex analysis reduces to a singleton.