12,542 research outputs found
On the Oß-hull of a planar point set
© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We study the Oß-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle ß. Given a set P of n points in the plane, we show how to maintain the Oß-hull of P while ß runs from 0 to p in T(n log n) time and O(n) space. With the same complexity, we also find the values of ß that maximize the area and the perimeter of the Oß-hull and, furthermore, we find the value of ß achieving the best fitting of the point set P with a two-joint chain of alternate interior angle ß.Peer ReviewedPostprint (author's final draft
Global surfaces of section for Reeb flows in dimension three and beyond
We survey some recent developments in the quest for global surfaces of
section for Reeb flows in dimension three using methods from Symplectic
Topology. We focus on applications to geometry, including existence of closed
geodesics and sharp systolic inequalities. Applications to topology and
celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written
for Proceedings of the ICM, Rio 2018; in v3 we made minor additional
corrections, updated references, added a reference to work of Lu on the
Conley Conjectur
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
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