Summary We characterize parallelepipeds in Rm within the family of all convex bodies by a property of special measures on its boundary. We show that these measures are related to weak derivatives (in the sense of  and ) of convex-valued functions. The results can be applied (see ) to derive a generalization of a theorem of Lehmann (see ) on the comparison of uniform location experiments.
Summary We generalize a result of Lehmann on the comparison of location experiments with uniform distributions on intervals. We compare a location experiment consisting of uniform distributions on parallelepipeds with a location experiment consisting of uniform distributions on convex bodies. We show that the first experiment can only be more informative than the second one if the convex bodies in the second experiment are themselves parallelepipeds. Further we show that the length of the edges of these parallelepipeds must fulfill a condition similar to the condition on the length of the intervals in Lehmann’s result.