Line transversals to disjoint balls

We prove that the set of directions of lines intersecting three disjoint balls in $R^3$ in a given order is a strictly convex subset of $S^2$. We then generalize this result to $n$ disjoint balls in $R^d$. 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

Helly-type Theorems for Line transversals to Disjoint Unit Balls (Extended abstract)

URL : http://delaunay.tem.uoc.gr/~mkaravel/ewcg06/papers/21.pdfInternational audienceWe prove Helly-type theorems for line transversals to disjoint unit balls in R^d. In particular, we show that a family of n >= 2d disjoint unit balls in Rd has a line transversal if, for some ordering of the balls, every subfamily of 2d balls admits a line transversal consistent with . We also prove that a family of n >= 4d − 1 disjoint unit balls in R^d admits a line transversal if every subfamily of size 4d − 1 admits a transversal

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

High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities

We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness''. The latter quantity adds up scaled errors of approximations of a measure by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors d-regular measure by its Jones-type flatness.Comment: 47 pages, 13 figures. Minor revisions and the inclusion of figure