When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2018.8
URN: urn:nbn:de:0030-drops-88343
URL: https://drops.dagstuhl.de/opus/volltexte/2018/8834/
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### Convex Hulls in Polygonal Domains

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### Abstract

We study generalizations of convex hulls to polygonal domains with holes. Convexity in Euclidean space is based on the notion of shortest paths, which are straight-line segments. In a polygonal domain, shortest paths are polygonal paths called geodesics. One possible generalization of convex hulls is based on the "rubber band" conception of the convex hull boundary as a shortest curve that encloses a given set of sites. However, it is NP-hard to compute such a curve in a general polygonal domain. Hence, we focus on a different, more direct generalization of convexity, where a set X is geodesically convex if it contains all geodesics between every pair of points x,y in X. The corresponding geodesic convex hull presents a few surprises, and turns out to behave quite differently compared to the classic Euclidean setting or to the geodesic hull inside a simple polygon. We describe a class of geometric objects that suffice to represent geodesic convex hulls of sets of sites, and characterize which such domains are geodesically convex. Using such a representation we present an algorithm to construct the geodesic convex hull of a set of O(n) sites in a polygonal domain with a total of n vertices and h holes in O(n^3h^{3+epsilon}) time, for any constant epsilon > 0.

### BibTeX - Entry

```@InProceedings{barba_et_al:LIPIcs:2018:8834,
author =	{Luis Barba and Michael Hoffmann and Matias Korman and Alexander Pilz},
title =	{{Convex Hulls in Polygonal Domains}},
booktitle =	{16th Scandinavian Symposium and Workshops on Algorithm  Theory (SWAT 2018)},
pages =	{8:1--8:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-068-2},
ISSN =	{1868-8969},
year =	{2018},
volume =	{101},
editor =	{David Eppstein},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},