7 research outputs found

    Line transversals to disjoint balls

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    We prove that the set of directions of lines intersecting three disjoint balls in R3R^3 in a given order is a strictly convex subset of S2S^2. We then generalize this result to nn disjoint balls in RdR^d. As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.Comment: 21 pages, includes figure

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    Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

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    International audienceWe prove Helly-type theorems for line transversals to disjoint unit balls in Rd\R^{d}. In particular, we show that a family of n≥2dn \geq 2d disjoint unit balls in Rd\R^d has a line transversal if, for some ordering ≺\prec of the balls, any subfamily of 2d2d balls admits a line transversal consistent with ≺\prec. We also prove that a family of n≥4d−1n \geq 4d-1 disjoint unit balls in Rd\R^d admits a line transversal if any subfamily of size 4d−14d-1 admits a transversal
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