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

TL;DR

This paper establishes a link between the Riemann integrability of a multifunction and its convex hull in Banach spaces, characterizing B-convexity through this integrability equivalence.

Contribution

It proves the equivalence of B-convexity of Banach spaces with the Riemann integrability correspondence between multifunctions and their convex hulls.

Findings

B-convexity characterized by integrability equivalence

Equivalence holds for multifunctions with compact values in any Banach space

Riemann integrability of convex hulls implies integrability of original multifunctions

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 $X$.