Adding to the answer by @Robert Israel (which I'm sure he knows, but didn't mention):

Because log is strictly monotonically increasing, this problem is equivalent to $maximize  \left( \|x\|_\infty \right)$ subject to the same constraints. 

That is a concave programming problem, i.e., the minimization of a concave function ($ -\|x\|_\infty$) subject to convex constraints. If the constraints are compact, there is a global optimum at an extreme of the constraints, although there may be additional global optima not at an extreme.  There may also be one or more local optima which are not global optima.