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A collection of k balls is independent if every subcollection has a point contained in all balls in the subcollection and not in any ball not in the subcollection. For balls in R3 there are independent collections for k = 1, 2, 3, 4. The main result of this work is the realization that complexes of independent collections correspond to short inclusion-exclusion formulas measuring the union of balls. A specific example is the dual complex defined by Edelsbrunner in 1995 whose simplices correspond to terms in an inclusion-exclusion formula. The new insight is an extension of this result to general independent complexes that have the same boundary. We also prove that the inclusion-exclusion formulas that correspond to such independent complexes are minimal.
The primary motivation for studying independent complexes is that they are significantly more stable (insensitive to motion) than the dual complex, which is a subcomplex of the Delaunay triangulation. This relative stability can perhaps be used to maintain geometric measures of a protein within a molecular dynamics loop in a small fraction of the time needed right now.