On the width and roundness of a set of points in the plane

Let S be a set of points in the plane. The width (resp. roundness) of S is defined as the minimum width of any slab (resp. annulus) that contains all points of S. We give a new characterization of the width of a point set. Al so, we give a rigorous proof of the fact that either the roundness of S is equal to the width of S, or the center of the minimum-width annulus is a ve rtex of the closest-point Voronoi diagram of S, the furthest-point Voronoi diagram of S, or an intersection point of these two diagrams. This proof co rrects the characterization of roundness used extensively in the literature .