Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums
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Approximation problems involving a single convex body in d-dimensional space have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to dimensions d ≤ 3 and/or do not consider approximation. In this paper, we consider approximations to two natural problems involving multiple convex bodies: detecting whether two polytopes intersect and computing their Minkowski sum. Given an approximation parameter ε > 0, we show how to independently preprocess two polytopes A,B into data structures of size O(1/ε^(d-1)/2) such that we can answer in polylogarithmic time whether A and B intersect approximately. More generally, we can answer this for the images of A and B under affine transformations. Next, we show how to ε-approximate the Minkowski sum of two given polytopes defined as the intersection of n halfspaces in O(n (1/ε) + 1/ε^(d-1)/2 + α) time, for any constant α > 0. Finally, we present a surprising impact of these results to a well studied problem that considers a single convex body. We show how to ε-approximate the width of a set of n points in O(n (1/ε) + 1/ε^(d-1)/2 + α) time, for any constant α > 0, a major improvement over the previous bound of roughly O(n + 1/ε^d-1) time.
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