The set of limits of Riemann integral sums of a multifunction and Banach space geometry

TL;DR

This paper investigates the geometric properties of the set of limit points of Riemann integral sums of multifunctions in Banach spaces, revealing convexity, star-shapedness, and conditions for emptiness.

Contribution

It establishes new results on the structure of limit sets of Riemann sums for multifunctions in Banach spaces, including convexity and star-shapedness conditions.

Findings

$I(F)$ is convex in finite-dimensional spaces.

$I(F)$ equals $I( ext{conv} F)$ in B-convex spaces or for compact-valued multifunctions.

$I(F)$ can be empty in infinite-dimensional Banach spaces.

Abstract

Let $X$ be a Banach space and $F: [0, 1] \rightarrow 2^{X} \setminus \{ \varnothing \}$ be a bounded multifunction. We study properties of the set $I(F)$ of limits in Hausdorff distance of Riemann integral sums of $F$. The main results are: (1) $I(F)$ is convex in the case of finite-dimensional $X$; (2) $I(F) = I(\operatorname{conv} F)$ in B-convex spaces or for compact-valued multifunctions; (3) $I(F)$ consists of convex sets whenever $X$ is B-convex; (4) $I(F)$ is star-shaped (thus non-empty) for compact-valued multifunctions in separable spaces. (5) For each infinite-dimensional Banach space there is a bounded multifunction with empty $I(F)$.