A Generalization of the maximal-spacings in several dimensions and a convexity test

The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (1983) for data uniformly distributed on the unit cube. Later on, Janson (1987) extended the results to data uniformly distributed on any bounded set, and obtained a very fine result, namely, he derived the asymptotic distribution of different maximal-spacings notions. These results have been very useful in many statistical applications. We extend Janson's results to the case where the data are generated from a H\"older continuous density that is bounded from below and whose support is bounded. As an application, we develop a convexity test for the support of a distribution.