Young's inequality for convolution and its applications in convex- and set-valued analysis

A version of Young's inequality for convolution is introduced and employed to some topics in convex- and set-valued analysis. The following problems are considered: uniform equivalence of metrics for convex subsets of the Euclidean space, the regularity of set-valued mappings and the continuity of the Funk–Radon transform. Also an isoperimetric inequality based on Young's inequality is introduced.