  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 $I(F)$.
