An empty subdifferential

Question

Can you give me an example of function $f$ defined on an Hilbert space, real valued (extended with $+ \infty$), lower semi continuous, convex and proper for which $\operatorname{dom}(\partial f)= \emptyset$?

For $\operatorname{dom}(\partial f)$ I mean the domain of subdifferentiability of $f$.

Answer 1

This is impossible. For a proper convex and lower semicontinuous function the domain of definition of the subdifferential is dense in the domain of $f$, $$ \mathrm{dom}(f) = \overline{ D(\partial f)}. $$ This implies that there is $x\in X$ such that $\partial f(x)\ne\emptyset$, as for proper convex and lower semicontinuous $f$ it holds $\mathrm{dom}(f) \ne \emptyset$.

This can be found for instance in Barbu/Precupanu: Convexity and optimization in Banach spaces, Corollary 2.44. There it is proven as a corollary to the result that the subdifferential is a maximal monotone operator (plus the theory of such mappings).

Answer 2

Under your assumptions the subdifferential $\partial f$ is a maximal monotone operator so it cannot be empty.