On the consistency of the spacings test for multivariate uniformity

We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set $K \subset \mathbb{R}^d$ that is based on the maximal spacing generated by i.i.d. points $X_1, \ldots,X_n$ in $K$, i.e., the volume of the largest convex set of a given shape that is contained in $K$ and avoids each of these points. Since asymptotic results for the case $d > 1$ are only availabe under uniformity, a key element of the proof is a suitable coupling.