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Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that each cell has the same nearest neighbors in S. Classically, both structures can be computed in O(n log n) time and O(n) space. We study the situation when the available… (More)

Let k be a nonnegative integer. In the approximate k-flat nearest neighbor (k-ANN) problem, we are given a set P ⊂ R d of n points in d-dimensional space and a fixed approximation factor c > 1. Our goal is to preprocess P so that we can efficiently answer approximate k-flat nearest neighbor queries: given a k-flat F , find a point in P whose distance to F… (More)

Let P 1 ,. .. , P d+1 ⊂ R d be d-dimensional point sets such that the convex hull of each P i contains the origin. We call the sets P i color classes, and we think of the points in P i as having color i. A colorful choice is a set with at most one point of each color. The colorful Carathéodory theorem guarantees the existence of a colorful choice whose… (More)

Let P be a d-dimensional n-point set. A partition T of P is called a Tverberg partition if the convex hulls of all sets in T intersect in at least one point. We say T is t-tolerated if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerated Tverberg partition with n/(d + 1)(t + 1)… (More)

Let C 1 , ..., C d+1 be d + 1 point sets in R d , each containing the origin in its convex hull. A subset C of d+1 i=1 C i is called a colorful choice (or rainbow) for C 1 ,. .. , C d+1 , if it contains exactly one point from each set C i. The colorful Carathéodory theorem states that there always exists a colorful choice for C 1 ,. .. , C d+1 that has the… (More)

Let P be a set of n points in d-dimensions. The simplicial depth, σ P (q) of a point q is the number of d-simplices with vertices in P that contain q in their convex hulls. The simplicial depth is a notion of data depth with many applications in robust statistics and computational geometry. Computing the simplicial depth of a point is known to be a… (More)

Let P be a d-dimensional n-point set. A partition T of P is called a Tverberg partition if the convex hulls of all sets in T intersect in at least one point. We say T is t-tolerated if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerated Tverberg partition with n/(d + 1)(t + 1)… (More)

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