## Advances in Applied Probability

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

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

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

#### Abstract

The problem of estimating an unknown compact convex set *K* in the plane,
from a sample (*X*_{1},···,*X*_{n})
of points independently and uniformly distributed over *K*, is considered. Let
*K*_{n} be the convex hull of the sample, Δ be the Hausdorff
distance, and Δ_{n} := Δ (*K*, *K*_{n}).
Under mild conditions, limit laws for Δ_{n} are obtained. We find
sequences (*a*_{n}), (*b*_{n}) such that

(Δ_{n} - *b*_{n})/*a*_{n} → 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

**Source**

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

**Dates**

First available: 21 October 2002

**Permanent link to this document**

http://projecteuclid.org/euclid.aap/1035228070

**Digital Object Identifier**

doi:10.1239/aap/1035228070

**Mathematical Reviews number (MathSciNet)**

MR1642840

**Zentralblatt MATH identifier**

0912.60022

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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

**Keywords**

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

#### Citation

Bräker, H.; Hsing, T.; Bingham, N. H. On the Hausdorff distance between a convex set and an interior random convex hull. Advances in Applied Probability 30 (1998), no. 2, 295--316. doi:10.1239/aap/1035228070. http://projecteuclid.org/euclid.aap/1035228070.