{"title":"关于立方体镶嵌的一个稳定性定理","authors":"J. Pach, P. Frankl","doi":"10.20382/jocg.v9i1a13","DOIUrl":null,"url":null,"abstract":"It is shown that if a $d$-dimensional cube is decomposed into $n$ cubes, the side lengths of which belong to the interval $(1 − \\frac{n}{1/d 1 +1} , 1]$, then $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"69 1","pages":"387-390"},"PeriodicalIF":0.0000,"publicationDate":"2018-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A stability theorem on cube tessellations\",\"authors\":\"J. Pach, P. Frankl\",\"doi\":\"10.20382/jocg.v9i1a13\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It is shown that if a $d$-dimensional cube is decomposed into $n$ cubes, the side lengths of which belong to the interval $(1 − \\\\frac{n}{1/d 1 +1} , 1]$, then $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.\",\"PeriodicalId\":54969,\"journal\":{\"name\":\"International Journal of Computational Geometry & Applications\",\"volume\":\"69 1\",\"pages\":\"387-390\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2018-07-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Computational Geometry & Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.20382/jocg.v9i1a13\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Computational Geometry & Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.20382/jocg.v9i1a13","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
It is shown that if a $d$-dimensional cube is decomposed into $n$ cubes, the side lengths of which belong to the interval $(1 − \frac{n}{1/d 1 +1} , 1]$, then $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.
期刊介绍：
The International Journal of Computational Geometry & Applications (IJCGA) is a quarterly journal devoted to the field of computational geometry within the framework of design and analysis of algorithms.
Emphasis is placed on the computational aspects of geometric problems that arise in various fields of science and engineering including computer-aided geometry design (CAGD), computer graphics, constructive solid geometry (CSG), operations research, pattern recognition, robotics, solid modelling, VLSI routing/layout, and others. Research contributions ranging from theoretical results in algorithm design — sequential or parallel, probabilistic or randomized algorithms — to applications in the above-mentioned areas are welcome. Research findings or experiences in the implementations of geometric algorithms, such as numerical stability, and papers with a geometric flavour related to algorithms or the application areas of computational geometry are also welcome.
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