Aperture-angle and Hausdorff-approximation of convex figures

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dc.contributor.authorAhn, HKko
dc.contributor.authorBae, Sko
dc.contributor.authorCheong, Otfriedko
dc.contributor.authorGudmundsson, Jko
dc.date.accessioned2008-10-30T08:44:14Z-
dc.date.available2008-10-30T08:44:14Z-
dc.date.created2012-02-06-
dc.date.created2012-02-06-
dc.date.issued2008-10-
dc.identifier.citationDISCRETE & COMPUTATIONAL GEOMETRY, v.40, no.3, pp.414 - 429-
dc.identifier.issn0179-5376-
dc.identifier.urihttp://hdl.handle.net/10203/7708-
dc.description.abstractThe aperture angle alpha(x, Q) of a point x is not an element of Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon Q. C is the minimum aperture angle of any x is an element of C \ Q with respect to Q. We show that for any compact convex set C in the plane and any k > 2, there is an inscribed convex k-gon Q subset of C with aperture angle approximation error (1 - 2/k+1)pi. This bound is optimal, and settles a conjecture by Fekete from the early 1990s. The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P) with Hausdorff distance sigma, but all subpolygons of P (the convex hull of some vertices of P) admit such an approximation, then P is a (k + 1)-gon. This implies the following result: For any k > 2 and any convex polygon P of perimeter at most 1 there is a sub-k-gon Q of P such that the Hausdorff-distance of P and Q is at most 1/k+1 sin pi/k+1.-
dc.description.sponsorshipThis research was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-311-D00763).en
dc.languageEnglish-
dc.language.isoen_USen
dc.publisherSPRINGER-
dc.titleAperture-angle and Hausdorff-approximation of convex figures-
dc.typeArticle-
dc.identifier.wosid000259563600008-
dc.identifier.scopusid2-s2.0-52949129795-
dc.type.rimsART-
dc.citation.volume40-
dc.citation.issue3-
dc.citation.beginningpage414-
dc.citation.endingpage429-
dc.citation.publicationnameDISCRETE & COMPUTATIONAL GEOMETRY-
dc.identifier.doi10.1007/s00454-007-9039-5-
dc.contributor.localauthorCheong, Otfried-
dc.contributor.nonIdAuthorAhn, HK-
dc.contributor.nonIdAuthorBae, S-
dc.contributor.nonIdAuthorGudmundsson, J-
dc.type.journalArticleArticle-
dc.subject.keywordAuthorHausdorff approximation-
dc.subject.keywordAuthoraperture angle-
dc.subject.keywordAuthorconvex figure-
dc.subject.keywordAuthorsubpolygon-
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