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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 $C$.

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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.

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

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