On the expected number of facets for the convex hull of samples
arxiv(2024)
Abstract
This paper studies the convex hull of d-dimensional samples i.i.d.
generated from spherically symmetric distributions. Specifically, we derive a
complete integration formula for the expected facet number of the convex hull.
This formula is with respect to the CDF of the radial distribution. As the
number of samples approaches infinity, the integration formula enables us to
obtain the asymptotic value of the expected facet number for three categories
of spherically symmetric distributions. Additionally, the asymptotic result can
be applied to estimating the sample complexity in order that the probability
measure of the convex hull tends to one.
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