In a random point cloud in R^{n}, say d_{1} <= d_{2} <= ... d_{k}
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
vol_{k} = k / d_{k}^{n} for a fixed k, not using d_{1} d_{2} ... at all.

For example, one could take a weighted average of the naive
vol_{1} vol_{2} ..., 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.)