A characterization of essentially strictly convex functions on reflexive Banach spaces

When some variational properties force convexity

“…4.5.2). In the same spirit, and for X reflexive, it has been recently proved ( [25], [Thm. 1) that a weakly lsc function J is essentially strictly convex if and only if J is adequate, a property we denote here by (A).…”

“…Condition (A) amounts to the notion of adequate function introduced in [25], for reflexive X. The main result about this notion is the following.…”

“…3.3, 3.9). The results presented here can also be applied to nearest and farthest point problems with respect to Bregman distances as in [25,27], a topic we do not tackle in this paper.…”

On inequalities for convex functions

“…Using that characterization, the authors are able to prove some non-trivial theorems concerning operator monotone functions. For some more recent characterizations, the interested reader can also consult the results in [3,5,8,9,14]. In this article we will give a property for convex functions that, similarly to midconvexity, gives almost a characterization (resulting an equivalent property in the situation of boundedness over one single interval).…”

Strongly adequate functions on Banach spaces

“…The notion of adequate function on a reflexive Banach space has been recently introduced to obtain a characterization for the class of essentially strictly convex functions (in the sense of [4]) among the weakly lower semicontinuous ones ([19,Th. 1]).…”