An alternative to the convexity of values in set-valued analysis

The convexity of the values of set-valued maps is a fundamental and standard assumption in set-valued analysis: indeed, it plays a crucial role in establishing fixed point theorems, selection results, the existence of equilibrium problems, etc. In this work, we propose an alternative approach based on a new concept, called Regular Local Convexity, that is proved to be sufficient to obtain extensions of the famous Michael’s selection theorem and other well-known results for the existence of suitable submap selections with desirable properties. As an application of our theoretical developments with this new concept, we analyze some classical mathematical problem arising from fixed points, economic theory, and differential inclusions.