11 research outputs found
There are arbitrary large minimal 2-pinning configurations
International audienceWe characterize minimal configurations that pin a line in every -plane
Lower Bounds for Pinning Lines by Balls
A line L is a transversal to a family F of convex objects in R^d if it
intersects every member of F. In this paper we show that for every integer d>2
there exists a family of 2d-1 pairwise disjoint unit balls in R^d with the
property that every subfamily of size 2d-2 admits a transversal, yet any line
misses at least one member of the family. This answers a question of Danzer
from 1957
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
Lines pinning lines
A line g is a transversal to a family F of convex polytopes in 3-dimensional
space if it intersects every member of F. If, in addition, g is an isolated
point of the space of line transversals to F, we say that F is a pinning of g.
We show that any minimal pinning of a line by convex polytopes such that no
face of a polytope is coplanar with the line has size at most eight. If, in
addition, the polytopes are disjoint, then it has size at most six. We
completely characterize configurations of disjoint polytopes that form minimal
pinnings of a line.Comment: 27 pages, 10 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
Bounding Helly numbers via Betti numbers
We show that very weak topological assumptions are enough to ensure the
existence of a Helly-type theorem. More precisely, we show that for any
non-negative integers and there exists an integer such that
the following holds. If is a finite family of subsets of such that for any
and every
then has Helly number at most . Here
denotes the reduced -Betti numbers (with singular homology). These
topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based
approach to build, given an arbitrary simplicial complex , some well-behaved
chain map .Comment: 29 pages, 8 figure
Hadwiger and Helly-type theorems for disjoint unit spheres in R³
Let S be an ordered set of disjoint unit spheres in R³. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal for S