The wikipedia says that $log-concavity$ is closed under products.
I am wondering how this translates to a function that is log concave over intervals but maybe not over the union of intervals.
My guess is that $f(x)g(x)$ would be log-concave over the respective integrals, but not over the union of the intervals as well. - is this correct?
My reasoning for this is that we could just apply the fact that log-concavity is closed under products to each of the intervals individually?(by intervals I mean the intervals where $Log(f(x))$ is concave) - the wikipedia doesn't say anything about the domains though, so I'm not 100% sure I'm correct about this. - (although I guess the proof using the fact that the log of the product is the sum of the logs should hold over each domain, so my question is probably trivial)