2,396 research outputs found
On the Complexity of the k-Level in Arrangements of Pseudoplanes
A classical open problem in combinatorial geometry is to obtain tight
asymptotic bounds on the maximum number of k-level vertices in an arrangement
of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes
passing below them). This is a dual version of the k-set problem, which, in a
primal setting, seeks bounds for the maximum number of k-sets determined by n
points in d dimensions, where a k-set is a subset of size k that can be
separated from its complement by a hyperplane. The k-set problem is still wide
open even in the plane, with a substantial gap between the best known upper and
lower bounds. The gap gets larger as the dimension grows. In three dimensions,
the best known upper bound is O(nk^(3/2)).
In its dual version, the problem can be generalized by replacing hyperplanes
by other families of surfaces (or curves in the planes). Reasonably sharp
bounds have been obtained for curves in the plane, but the known upper bounds
are rather weak for more general surfaces, already in three dimensions, except
for the case of triangles. The best known general bound, due to Chan is
O(n^2.997), for families of surfaces that satisfy certain (fairly weak)
properties.
In this paper we consider the case of pseudoplanes in 3 dimensions (defined
in detail in the introduction), and establish the upper bound O(nk^(5/3)) for
the number of k-level vertices in an arrangement of n pseudoplanes. The bound
is obtained by establishing suitable (and nontrivial) extensions of dual
versions of classical tools that have been used in studying the primal k-set
problem, such as the Lova'sz Lemma and the Crossing Lemma.Comment: 23 pages, 13 figure
On Ray Shooting for Triangles in 3-Space and Related Problems
We consider several problems that involve lines in three dimensions, and
present improved algorithms for solving them. The problems include (i) ray
shooting amid triangles in , (ii) reporting intersections between query
lines (segments, or rays) and input triangles, as well as approximately
counting the number of such intersections, (iii) computing the intersection of
two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections
in a set of lines in , and (v) output-sensitive construction of an
arrangement of triangles in three dimensions.
Our approach is based on the polynomial partitioning technique.
For example, our ray-shooting algorithm processes a set of triangles in
into a data structure for answering ray shooting queries amid the given
triangles, which uses storage and preprocessing, and
answers a query in time, for any . This
is a significant improvement over known results, obtained more than 25 years
ago, in which, with this amount of storage, the query time bound is roughly
. The algorithms for the other problems have similar performance
bounds, with similar improvements over previous results.
We also derive a nontrivial improved tradeoff between storage and query time.
Using it, we obtain algorithms that answer queries on objects in time, for any
, again an improvement over the earlier bounds.Comment: 33 pages, 7 figure
Experimental study of energy-minimizing point configurations on spheres
In this paper we report on massive computer experiments aimed at finding
spherical point configurations that minimize potential energy. We present
experimental evidence for two new universal optima (consisting of 40 points in
10 dimensions and 64 points in 14 dimensions), as well as evidence that there
are no others with at most 64 points. We also describe several other new
polytopes, and we present new geometrical descriptions of some of the known
universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic
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