Non-subdifferentiable convex function on a normed space

Question

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?

Answer 1

From Rockafellar's Convex Optimization book, Theorem 23.3:

Let $f$ be a convex function, and let $x$ be a point where $f$ is finite. If $f$ is subdifferentiable at $x$, then it is proper. If $f$ is not subdifferentiable at $x$, there must be some infinite two-sided directional derivative at $x$, i.e. there must exist some $y$ such that $f'(x;y) = -f'(x,-y) = -\infty$.

Additionally, Rockafellar Theorem 23.4 says that if $f$ is a proper convex function, then if $x \in \text{ri} (\text{dom} f)$ then $\partial f(x)$ is non-empty.

Or in other words, if you have a proper convex function with a non-empty relative interior, then there is at least one point where it is subdifferentiable.

To construct an example of a convex function that is never subdifferentiable, we can build a function that has no relative interior and whose directional derivative in its domain is always infinite in at least one direction. I believe this is one such example: $$ f(x) = \begin{cases} -\infty & \text{if } ||x||<1, \\ 0 & \text{if } ||x|| =1, \\ +\infty & \text{if } ||x||>1 \end{cases} $$

Answer 2

A convex function must be subdifferentiable on the relative interior of its domain (where it is finite). So the only points where one can have an issue is the boundary of its domain.

The function constructed by Kevin Holt is one where all of the domain is boundary--hence Theorem 23.4 in Rockafellar's Convex Analysis cannot be used.

As another illustration of how things can go badly at boundary points, consider the function $$g(x) = \left\{ \begin{array}{cc} 1-\sqrt{x} & x \geq 0 \\ \infty & \text{otherwise} \end{array} \right.$$ which is convex and defined at $0$, but not subdifferentiable there. A linear equation tangent to $g$ at $0$ must have infinite slope.