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Separable Partitions Shmuel Onn

Summary: Separable Partitions
Noga Alon
Shmuel Onn
An ordered partition of a set of n points in the d dimensional Euclidean space is called
a separable partition if the convex hulls of the parts are pairwise disjoint. For each fixed
p and d we determine the maximum possible number rp,d(n) of separable partitions into
p parts of n points in real d-space up to a constant factor. Of particular interest are
the values rp,d(n) = (nd(p
2)) for every fixed p and d 3, and rp,2(n) = (n6p-12
) for
every fixed p 3. We establish similar results for spaces of finite Vapnik-Chervonenkis
dimension and study the corresponding problem for points on the moment curve as well.
1 Introduction
A separable p-partition of a set of n points in the d-dimensional Euclidean space Rd is an
ordered tuple = (1, . . . , p) of p nonempty sets whose disjoint union is S, where the convex
hulls of the sets j are pairwise disjoint. Let rp,d(n) denote the maximum possible number
of separable p partitions of a set of n points in Rd. It is easy to see that rp,1(n) = p! n-1
p-1 =
(np-1). The following theorem determines the asymptotic behavior of rp,d(n) for every


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics