On intersecting a point set with Euclidean balls

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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.




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

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