Which are the most famous problems having an objective of maximizing a nonlinear convex function (or minimizing a concave function)? As far as I know such an objective with respect to linear constraints is np-hard.
THE most famous problem having an objective of maximizing a convex function (or minimizing a concave function), and having linear constraints, is Linear Programming, which is NOT np-hard.
Linear Programming is both a convex optimization problem (minimizing a convex function subject to convex constraints) and a concave optimization problem (minimizing a concave function subject to convex constraints). Therefore it has all the properties of both, to include, all local optima are global optima, and if the constraints are compact (bounded), then there is a global optimum at the extreme of the constraints.
Regarding convex maximization problems, I would recommend the paper by R.Enkhbat " Global Optimization approach to Malfatti's problem"(Journal of Global Optimization, 2016). I think Malfatti's problem which was formulated in 1803 will be a nice example of the convex maximization problem.