7 research outputs found
Line transversals to disjoint balls
We prove that the set of directions of lines intersecting three disjoint
balls in in a given order is a strictly convex subset of . We then
generalize this result to disjoint balls in . 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
Helly-Type Theorems for Line Transversals to Disjoint Unit Balls
International audienceWe prove Helly-type theorems for line transversals to disjoint unit balls in . In particular, we show that a family of disjoint unit balls in has a line transversal if, for some ordering of the balls, any subfamily of balls admits a line transversal consistent with . We also prove that a family of disjoint unit balls in admits a line transversal if any subfamily of size admits a transversal