Random convex hulls: floating bodies and expectations

Abstract Recently Barany and Larman have shown that the volume of the convex hull of a point sample taken uniformly within a convex body can be approximated by the volume of a related "floating body." Here we show that, in the sense of sets, the floating body and the expectation of the convex hull are close. As an immediate consequence of the floating body as intermediary, we observe that the expectation of the volume of the convex hull is approximately the same as the volume of its expectation, an issue related to the Brunn-Minkowski inequality.