Does a log-concave function on a convex set extend continuously to the boundary? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:50:00Z http://mathoverflow.net/feeds/question/47246 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47246/does-a-log-concave-function-on-a-convex-set-extend-continuously-to-the-boundary Does a log-concave function on a convex set extend continuously to the boundary? Tom LaGatta 2010-11-24T16:16:01Z 2010-11-25T07:45:58Z <p>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 $\overline U$?</p> http://mathoverflow.net/questions/47246/does-a-log-concave-function-on-a-convex-set-extend-continuously-to-the-boundary/47305#47305 Answer by Theo Buehler for Does a log-concave function on a convex set extend continuously to the boundary? Theo Buehler 2010-11-25T07:45:58Z 2010-11-25T07:45:58Z <p>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.</p> <p>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.</p>