Minimal 2-Fold Coverings of E d

Abstract

Let N(k, d) be the smallest positive integer such that given any finite collection of open halfspaces which k-fold coversEd, there exists a subcollection of cardinality less than or equal toN(k,d) which k-fold coversEd. A well-known corollary to Helly's theorem proves N(1,d) =d+1. This provides an inductive base from which we show N(k; d) exists for all positive integers k.Our main result is \(N(2,d) = \left\lfloor {((d + 3)/2)^2 } \right\rfloor \).