Vertical Developments of a Convex Function

In this paper we compare two different approaches to analyse the second-order behaviour of a convex function. The first one is classical, we call it the horizontal approach; the second one is more recent, it is the vertical approach. We prove equivalences between horizontal and vertical growth conditions. Then we derive well-known directional results. Finally we show that the vertical approach is particularly interesting to get more than a first-order (and more than directional) analysis of the maximum eigenvalue function.