Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

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

Inflating balls is NP-hard

à paraîtreInternational audienceA collection C of balls in R^d is \delta-inflatable if it is isometric to the intersection U \cap E of some d-dimensional affine subspace E with a collection U of (d+\delta)-dimensional balls that are disjoint and have equal radius. We give a quadratic-time algorithm to recognize 1-inflatable collections of balls in any fixed dimension, and show that recognizing \delta-inflatable collections of d-dimensional balls is NP-hard for \delta \geq 2 and d \geq 3 if the balls' centers and radii are given by numbers of the form a+b\sqrt{c+d\sqrt{e}}, where a, ..., e are integers