In a random point cloud in Rn, say d1 <= d2 <= ... dk are the distances of the k points nearest the origin — I know only their distances, not their coordinates. What's a "good" estimate of the point density near 0 ?
In density estimation in statistics, it seems common to use
volk = k / dkn for a fixed k, not using d1 d2 ... at all.
For example, one could take a weighted average of the naive vol1 vol2 ..., but with what weights ?
Added, re convex hull: can't say — 0 might not even be in the convex hull (cf. Wendel's formula). So there are two related but separate questions: estimating volume (answered by Joseph O'Rourke's reference), and estimating density.
(Experts, please add tags.)