{"title":"关于广义闵可夫斯基排列","authors":"M'at'e Kadlicsk'o, Z. L'angi","doi":"10.26493/1855-3974.2550.d96","DOIUrl":null,"url":null,"abstract":"The concept of a Minkowski arrangement was introduced by Fejes T\\'oth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes T\\'oth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $\\mu$ for some $0<\\mu<1$ if no member $K$ of the family overlaps the homothetic copy of any other member $K'$ with ratio $\\mu$ and with the same center as $K'$. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order $\\mu$ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes T\\'oth for Minkowski arrangements of circular disks, and a result of B\\\"or\\\"oczky and Szab\\'o about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On generalized Minkowski arrangements\",\"authors\":\"M'at'e Kadlicsk'o, Z. L'angi\",\"doi\":\"10.26493/1855-3974.2550.d96\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The concept of a Minkowski arrangement was introduced by Fejes T\\\\'oth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes T\\\\'oth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $\\\\mu$ for some $0<\\\\mu<1$ if no member $K$ of the family overlaps the homothetic copy of any other member $K'$ with ratio $\\\\mu$ and with the same center as $K'$. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order $\\\\mu$ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes T\\\\'oth for Minkowski arrangements of circular disks, and a result of B\\\\\\\"or\\\\\\\"oczky and Szab\\\\'o about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.\",\"PeriodicalId\":8402,\"journal\":{\"name\":\"Ars Math. Contemp.\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-02-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ars Math. Contemp.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.26493/1855-3974.2550.d96\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Math. Contemp.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/1855-3974.2550.d96","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
摘要
闵可夫斯基排列的概念是由Fejes T\ oth在1965年引入的,它是一组中心对称的凸体,其性质是该族中的任何成员都不包含其内部任何其他成员的中心。这个概念在1967年由Fejes T 'oth推广,他将中心对称凸体族称为阶$\mu$的广义闵可夫斯基排列,对于某些$0<\mu<1$,如果族中没有成员$K$与任何其他成员$K'$的同构拷贝以比值$\mu$重叠且中心与$K'$相同。本文证明了欧几里得平面上有限多个圆盘的广义Minkowski阶$\mu$排列的单元总面积的一个明显上界。这个结果是Fejes T 'oth关于圆盘的Minkowski排列的一个类似结果的一般推广,以及B ' or ' oczky和Szab 'o关于平面上圆盘的广义Minkowski排列的最大密度的一个结果。此外,我们给出了中心对称凸体的齐次副本的广义Minkowski排列的密度的一个尖锐上界。
The concept of a Minkowski arrangement was introduced by Fejes T\'oth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes T\'oth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $\mu$ for some $0<\mu<1$ if no member $K$ of the family overlaps the homothetic copy of any other member $K'$ with ratio $\mu$ and with the same center as $K'$. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order $\mu$ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes T\'oth for Minkowski arrangements of circular disks, and a result of B\"or\"oczky and Szab\'o about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.