Does a log-concave function on a convex set extend continuously to the boundary?

2010-11-24T16:16:01Z

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a continuous extension to the closure $\overline U$?

Answer by Theo Buehler for Does a log-concave function on a convex set extend continuously to the boundary?

2010-11-25T07:45:58Z

I do not see how $\log$-concavity should imply any form of continuity. For instance, if $\|\cdot\|$ is any semi-norm on the locally convex space $X$ then $f(x) = e^{-\|x\|}$ will be bounded and $\log$-concave but it will only be continuous if the semi-norm is continuous.

What you really want in order to be able to extend $f$ to the boundary of $U$ is uniform continuity and for that the fact that $f$ is $\log$-concave may help but is certainly not sufficient.

Does a log-concave function on a convex set extend continuously to the boundary? - MathOverflow

Does a log-concave function on a convex set extend continuously to the boundary?

Answer by Theo Buehler for Does a log-concave function on a convex set extend continuously to the boundary?