  We discuss, on finite and infinite dimensional normed vector spaces, some
versions of Radstr\"{o}m cancellation law (or lemma) that are suited for
applications to set optimization problems. In this sense, we call our results
"conic" variants of the celebrated result of Radstr\"{o}m, since they involve
the presence of an ordering cone on the underlying space. Several adaptations
to this context of some topological properties of sets are studied and some
applications to subdifferential calculus associated to set-valued maps and to
necessary optimality conditions for constrained set optimization problems are
given. Finally, a stability problem is considered.
