So I know that sum of log-concave functions need not be log-concave. But I was wondering if the sum of a strictly concave function with a log-concave function is log-concave. I am considering functions that a sum of a sigmoid and a norm. Something like this:
$$ \tfrac{e^x}{1+e^x} - (x-c)^2 $$
When I plot this, visually this looks log-concave. But can't find a more structured way to reason about this.
To make the search for a counterexample more systematic, consider the second derivative of the logarithm:
$$ (\log f)'=\frac{f'}f\;,\\ (\log f)''=\frac{ff''-f'^2}{f^2}\;. $$
So a function can be log-concave without being concave because the $-f'^2$ term can make it so despite $f$ and $f''$ being positive. That suggests that for a counterexample we need to increase the first term without appreciably changing the second. One way to do that is to just add a constant to increase $f$. For instance,
$$\frac{\mathrm e^x}{1+\mathrm e^x}+10$$
isn’t log-concave. Now just add a small convex part; for instance,
$$\frac{\mathrm e^x}{1+\mathrm e^x}+10-\frac{x^2}{100}$$
isn’t log-concave, but it’s the sum of a log-concave and a strictly concave function.