313 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 (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 . 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
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
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