83 research outputs found
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
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 numbers of Algebraic Subsets of
We study -convex sets, which are the geometric objects obtained as the
intersection of the usual convex sets in with a proper subset
. We contribute new results about their -Helly
numbers. We extend prior work for , , and ; we give sharp bounds on the -Helly numbers in
several new cases. We considered the situation for low-dimensional and for
sets that have some algebraic structure, in particular when is an
arbitrary subgroup of or when is the difference between a
lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz
method we obtain colorful versions of many monochromatic Helly-type results,
including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was
originally the first half of arXiv:1504.00076v
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
Geometric Permutations of Non-Overlapping Unit Balls Revisited
Given four congruent balls in that have disjoint
interior and admit a line that intersects them in the order , we show
that the distance between the centers of consecutive balls is smaller than the
distance between the centers of and . This allows us to give a new short
proof that interior-disjoint congruent balls admit at most three geometric
permutations, two if . We also make a conjecture that would imply that
such balls admit at most two geometric permutations, and show that if
the conjecture is false, then there is a counter-example of a highly degenerate
nature
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