Random approximation of convex bodies in Hausdorff metric
arxiv(2024)
摘要
While there is extensive literature on approximation, deterministic as well
as random, of general convex bodies K in the symmetric difference metric, or
other metrics arising from intrinsic volumes, very little is known for
corresponding random results in the Hausdorff distance when the approximant
K_n is given by the convex hull of n independent random points chosen
uniformly on the boundary or in the interior of K. When K is a polygon and
the points are chosen on its boundary, we determine the exact limiting behavior
of the expected Hausdorff distance between a polygon as n→∞. From this
we derive the behavior of the asymptotic constant for a regular polygon in the
number of vertices.
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