Univariate statistical theories have always inspired development of similar multivariate statistical theories. With the acceleration of computational capabilities in the last decade, dramatic progress in developing multivariate statistical methods has been made by virtue of high-speed computers and excellent software including S-Plus, Matlab, and libraries like IMSL and NAG.

Following univariate methods, many methods were generalized to develop multivariate density estimator. In two dimensions, histogram-type estimators developed on hexagonal bins were proved to give slightly better estimates in terms of Asymptotic Mean Integrated Squared Error (cf. Scott, 1988). But, this poses a question: what will be the counterpart of a hexagon in a dimension higher than two? The hexagon is the "roundest" of the space filling tessellation in two dimensions. In three dimensions, truncated tetrahedron has the roundest space filling tesselllation property. However, the analog in higher dimensions is unclear. An obvious, but suboptimal choice is tessellation via hyper-rectangles. The number of hyper-rectangular bins grows exponentially with dimension and pose the problem of the "Empty Space phenomenon," Scott and Thompson (1983) or "Curse of Dimensionality," Bellman (1961).

This problem inspired a different kind of binning: minimal space-filling simplicial tiles generated by the Delaunay tessellation. In this paper, complete algorithms and computational aspects are developed. Following this, we show that the estimator is consistent, is conditionally maximum likelihood, and we construct its asymptotic distribution. Also, we demonstrate empirically that the number of tiles grow much more slowly than with rectangular binning.