Higher Connectedness Properties of Support Points and Functionals of Convex Sets We prove that the set of all support points of a nonempty closed convex bounded set $C$ in a real infinite-dimensional Banach space $X$ is $\mathrm{AR(}\sigma$-$\mathrm{compact)}$ and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph and the range of the subdifferential map of a proper convex l.s.c. function on $X$. Keywords:convex set, support point, support functional, absolute retract, Leray-Schauder continuation principleCategories:46A55, 46B99, 52A07