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arXiv · 2105.10681

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

Vladimir Kadets·Artur Kulykov·Olha Shevchenko
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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 $X$.

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Vladimir Kadets, Artur Kulykov, Olha Shevchenko. 2021-05-22. Connection between the Riemann integrability of a multi-valued function and of its convex hull. https://doi.org/10.1016/j.jmaa.2021.125652

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