Advances in Applied Probability

On the Hausdorff distance between a convex set and an interior random convex hull

H. Bräker, T. Hsing, and N. H. Bingham

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The problem of estimating an unknown compact convex set K in the plane, from a sample (X1,···,Xn) of points independently and uniformly distributed over K, is considered. Let Kn be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that

n - bn)/an → Λ (n → ∞),

where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

Article information

Adv. in Appl. Probab. Volume 30, Number 2 (1998), 295-316.

First available in Project Euclid: 21 October 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]

Convex set convex hull Hausdorff metric limit law Gumbel distribution extreme value theory smooth boundary polygon moving boundary home range


Bräker, H.; Hsing, T.; Bingham, N. H. On the Hausdorff distance between a convex set and an interior random convex hull. Adv. in Appl. Probab. 30 (1998), no. 2, 295--316. doi:10.1239/aap/1035228070.

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