Connection between the Riemann integrability of a multi-valued function and of its convex hull

Abstract

For a Banach space X we demonstrate the equivalence of the following two properties:(1) X is B-convex (that is, possesses a nontrivial infratype), and(2) if F:[0,1]→2X∖{∅} is a multifunction with bounded values, convF denotes the multifunction t↦conv(F(t)), then the Riemann integrability of convF is equivalent to the Riemann integrability of F.For multifunctions with relatively norm compact values the Riemann integrability of convF is equivalent to the Riemann integrability of F without any restrictions on the Banach space X.