(Convex Hulls are Nonoverlapping) Consider a facility location instance with

nodes in R

2

and Euclidean distances. Suppose we open a set J

′ ⊆ J of facilities and assign

each customer in I to the nearest open facility. Recall that the neighborhood of an open

facility j is Nj ≡ {i ∈ I|yij=1}. Prove that the convex hulls of the neighborhoods of the

open facilities do not overlap.

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