On intersecting a point set with Euclidean balls

1Citations
Citations of this article
7Readers
Mendeley users who have this article in their library.

Abstract

The growth function for a class of subsets C of a set X is defined by mc(N) ≡ max { Δc(F): F ⊆ X, |F| = N} , N = l,2, ... , where Δc(F) ≡ |{F ∩ C: C ∈ C}|, the number of possible sets obtained by intersecting an element of C with the set F. Sauer (1972) showed that if C forms a Vapnik-Chervonenkis class with dimension V (C), then mc(N) ≤ j=0v(c)-1Σ (jN) for N ≥ V (C) - 1 The collection C of Euclidean balls in Rd has been shown by Dudley (1979) to have VC dimension equal to d + 2. It is well known, by using a standard geometric transformation, that Sauer's bound gives the exact number of subsets in this case. We give a more direct construction of the subsets picked out by balls, and as a corollary we obtain the number of such subsets.

Cite

CITATION STYLE

APA

Naiman, D. Q., & Wynn, H. P. (1997). On intersecting a point set with Euclidean balls. Computational Geometry: Theory and Applications, 7(4), 237–244. https://doi.org/10.1016/0925-7721(95)00037-2

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free