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 $F:[0,1]\to 2^X\setminus \{\varnothing \}$ is a {multifunction}, $\mathrm{conv}F$ denotes the mapping $t\mapsto \mathrm{conv}F(t)$, then the Riemann integrability of $\mathrm{conv}F$ is equivalent to the Riemann integrability of $F$. For multifunctions with compact values the Riemann integrability of $\mathrm{conv}F$ is equivalent to the Riemann integrability of $F$ without any restrictions on the Banach space $X$.