# Functional characterization of the boundary of a convex set

I have a closed convex set $C$ with non-empty interior in a Banach space X.

I try to find a functional characterization of the boundary of $C$ in the sense that I would like to associate to $C$ a convex function $f$ such that the boundary of $C$ is perfectly described in terms of $f$ and such that $f$ is at least lower semi-continuos ($C$ is closed just for that).

Of course I considered the indicator function of $C$ which is given by $\iota_C(x)=0$ if $x\in C$ and $\iota_C(x)=+\infty$ for $x\not\in C$ but this function cannot describe the boundary of $C$.

You can use $$f(x) = \begin{cases} +\infty & \text{if } x \not\in C \\ 1 & \text{if } x \in \partial C \\ 0 & \text{if } x \in \operatorname{int}(C)\end{cases}$$

Is it easy to check that this function is convex.

• Your answer helped me clarify what I want. I modified above. Nov 4, 2015 at 15:40
• Maybe the signed distance function does the job?
– gerw
Nov 4, 2015 at 18:23
• What do you mean by signed distance function? Nov 4, 2015 at 18:38
• en.wikipedia.org/wiki/Signed_distance_function But I am not sure, whether it works in an arbitrary Banach space.
– gerw
Nov 4, 2015 at 18:48
• I don't think that function is convex. Nov 4, 2015 at 19:01