David E. Hong, Dan Ismailescu, Alex Kwak, Grace Y. Park
{"title":"On the smallest area $$(n-1)$$-gon containing a convex n-gon","authors":"David E. Hong, Dan Ismailescu, Alex Kwak, Grace Y. Park","doi":"10.1007/s10998-023-00527-4","DOIUrl":null,"url":null,"abstract":"Approximation of convex disks by inscribed and circumscribed polygons is a classical geometric problem whose study is motivated by various applications in robotics and computer aided design. We consider the following optimization problem: given integers $$3\\le n\\le m-1$$ , find the value or an estimate of $$\\begin{aligned} r(n,m)=\\max _{P\\in {\\mathcal {P}}_m}\\,\\, \\min _{Q\\in {\\mathcal {P}}_n,\\,Q \\supseteq P} \\frac{|Q|}{|P|} \\end{aligned}$$ where P varies in the set $${\\mathcal {P}}_m$$ of all convex m-gons, and, for a fixed m-gon P, the minimum is taken over all n-gons Q containing P; here $$|\\cdot |$$ denotes area. It is easy to prove that $$r(3,4)=2$$ , and from a result of Gronchi and Longinetti it follows that $$r(n-1, n)= 1+\\frac{1}{n}\\tan \\left( \\pi /{n}\\right) \\tan \\left( {2\\pi }/{n}\\right) $$ for all $$n\\ge 6$$ . In this paper we show that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than $$3/\\sqrt{5}$$ thus determining the value of r(4, 5). In all cases, the equality is reached only for affine regular polygons.","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":"1 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Periodica Mathematica Hungarica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s10998-023-00527-4","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Approximation of convex disks by inscribed and circumscribed polygons is a classical geometric problem whose study is motivated by various applications in robotics and computer aided design. We consider the following optimization problem: given integers $$3\le n\le m-1$$ , find the value or an estimate of $$\begin{aligned} r(n,m)=\max _{P\in {\mathcal {P}}_m}\,\, \min _{Q\in {\mathcal {P}}_n,\,Q \supseteq P} \frac{|Q|}{|P|} \end{aligned}$$ where P varies in the set $${\mathcal {P}}_m$$ of all convex m-gons, and, for a fixed m-gon P, the minimum is taken over all n-gons Q containing P; here $$|\cdot |$$ denotes area. It is easy to prove that $$r(3,4)=2$$ , and from a result of Gronchi and Longinetti it follows that $$r(n-1, n)= 1+\frac{1}{n}\tan \left( \pi /{n}\right) \tan \left( {2\pi }/{n}\right) $$ for all $$n\ge 6$$ . In this paper we show that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than $$3/\sqrt{5}$$ thus determining the value of r(4, 5). In all cases, the equality is reached only for affine regular polygons.
期刊介绍:
Periodica Mathematica Hungarica is devoted to publishing research articles in all areas of pure and applied mathematics as well as theoretical computer science. To be published in the Periodica, a paper must be correct, new, and significant. Very strong submissions (upon the consent of the author) will be redirected to Acta Mathematica Hungarica.
Periodica Mathematica Hungarica is the journal of the Hungarian Mathematical Society (János Bolyai Mathematical Society). The main profile of the journal is in pure mathematics, being open to applied mathematical papers with significant mathematical content.