1,299 research outputs found
Halving Balls in Deterministic Linear Time
Let \D be a set of pairwise disjoint unit balls in and the
set of their center points. A hyperplane \Hy is an \emph{-separator} for
\D if each closed halfspace bounded by \Hy contains at least points
from . This generalizes the notion of halving hyperplanes, which correspond
to -separators. The analogous notion for point sets has been well studied.
Separators have various applications, for instance, in divide-and-conquer
schemes. In such a scheme any ball that is intersected by the separating
hyperplane may still interact with both sides of the partition. Therefore it is
desirable that the separating hyperplane intersects a small number of balls
only. We present three deterministic algorithms to bisect or approximately
bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present
a simple linear-time algorithm to construct an -separator for balls
in , for any , that intersects at most
balls, for some constant that depends on and . The number of
intersected balls is best possible up to the constant . Secondly, we present
a near-linear time algorithm to construct an -separator in
that intersects balls. Finally, we give a linear-time algorithm to
construct a halving line in that intersects
disks.
Our results improve the runtime of a disk sliding algorithm by Bereg,
Dumitrescu and Pach. In addition, our results improve and derandomize an
algorithm to construct a space decomposition used by L{\"o}ffler and Mulzer to
construct an onion (convex layer) decomposition for imprecise points (any point
resides at an unknown location within a given disk)
Cutting arcs for torus links and trees
Among all torus links, we characterise those arising as links of simple plane
curve singularities by the property that their fibre surfaces admit only a
finite number of cutting arcs that preserve fibredness. The same property
allows a characterisation of Coxeter-Dynkin trees (i.e., , , ,
and ) among all positive tree-like Hopf plumbings.Comment: 27 pages, 18 figures. Results have been extended to cover all
Coxeter-Dynkin trees in the new versio
Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone
By using a suitable triple cover we show how to possibly model the
construction of a minimal surface with positive genus spanning all six edges of
a tetrahedron, working in the space of BV functions and interpreting the film
as the boundary of a Caccioppoli set in the covering space. After a question
raised by R. Hardt in the late 1980's, it seems common opinion that an
area-minimizing surface of this sort does not exist for a regular tetrahedron,
although a proof of this fact is still missing. In this paper we show that
there exists a surface of positive genus spanning the boundary of an elongated
tetrahedron and having area strictly less than the area of the conic surface.Comment: Expanding on the previous version with additional lower bounds, new
images, corrections and improvements. Comparison with Reifenberg approac
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