219 research outputs found
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
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
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
Computational Geometry Column 41
The recent result that n congruent balls in R^d have at most 4 distinct
geometric permutations is described.Comment: To appear in SIGACT News and in Internat. J. Comput. Geom. App
Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents
This paper concerns the dynamics of polynomial automorphisms of .
One can associate to such an automorphism two currents and the
equilibrium measure . In this paper we study some
geometric and dynamical properties of these objects. First, we characterize
as the unique measure of maximal entropy. Then we show that the measure
has a local product structure and that the currents have a
laminar structure. This allows us to deduce information about periodic points
and heteroclinic intersections. For example, we prove that the support of
coincides with the closure of the set of saddle points. The methods used
combine the pluripotential theory with the theory of non-uniformly hyperbolic
dynamical systems
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