I'm looking at the Boyd & Vandenberghe slides on Convex Optimization. In slide 18, it applies the rules of vector composition on an example to say that it is convex. The example given is
$\log\sum_{i=1}^{m}\exp g_i$ is convex if $g_i$ is convex.
I know the exponential (and sum of exponential) function is convex. But isn't the log function concave? How does this fit the rule, since $h$ (I'm assuming $h$ is the log function) isn't convex?
A function $f$ is called log-convex if $\ln f$ is convex. It is not that difficult to show that a sum of two log-convex functions is log-convex. All you need to do is to notice that the function $\exp g_i$ is log-convex.
Another approach would be to show by definition for the case $m=2$ and then generalise to an arbitrary $m$.