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estimate volume inside density of k points in Rn from their distances ?

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 volume inside k points (in their convex hull), for uniformly distibuted points 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.

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show/hide this revision's text 2 convex hull

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 volume inside k points (in their convex hull), for uniformly distibuted points ?

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 ?

(Experts, please add tags.)
show/hide this revision's text 1

estimate volume inside k points in Rn from their distances ?

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 volume inside k points, for uniformly distibuted points ?

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 ?

(Experts, please add tags.)