  In this paper, we present a novel concept of the Fenchel conjugate for
set-valued mappings and investigate its properties in finite and infinite
dimensions. After establishing the fundamental properties of the Fenchel
conjugate for set-valued mappings, we derive its main calculus rules in various
settings. Our approach is geometric and draws inspiration from the successful
application of this method by B. S. Mordukhovich and coauthors in variational
and convex analysis. Subsequently, we demonstrate that our new findings for the
Fenchel conjugate of set-valued mappings can be utilized to obtain many old and
new calculus rules of convex generalized differentiation in both finite and
infinite dimensions.
