Asymmetric Convex Intersection Testing
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We consider asymmetric convex intersection testing (ACIT). Let P ⊂R^d be a set of n points and H a set of n halfspaces in d dimensions. We denote by ch(P) the polytope obtained by taking the convex hull of P, and by fh(H) the polytope obtained by taking the intersection of the halfspaces in H. Our goal is to decide whether the intersection of H and the convex hull of P are disjoint. Even though ACIT is a natural variant of classic LP-type problems that have been studied at length in the literature, and despite its applications in the analysis of high-dimensional data sets, it appears that the problem has not been studied before. We discuss how known approaches can be used to attack the ACIT problem, and we provide a very simple strategy that leads to a deterministic algorithm, linear on n and m, whose running time depends reasonably on the dimension d.
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