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Examples of tangent cones of non-collapsed Ricci limit spaces 非折叠利玛窦极限空间切锥实例
arXiv - MATH - Metric Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.11954
Philipp Reiser
{"title":"Examples of tangent cones of non-collapsed Ricci limit spaces","authors":"Philipp Reiser","doi":"arxiv-2409.11954","DOIUrl":"https://doi.org/arxiv-2409.11954","url":null,"abstract":"We give new examples of manifolds that appear as cross sections of tangent\u0000cones of non-collapsed Ricci limit spaces. It was shown by Colding-Naber that\u0000the homeomorphism types of the tangent cones of a fixed point of such a space\u0000do not need to be unique. In fact, they constructed an example in dimension 5\u0000where two different homeomorphism types appear at the same point. In this note,\u0000we extend this result and construct limit spaces in all dimensions at least 5\u0000where any finite collection of manifolds that admit core metrics, a type of\u0000metric introduced by Perelman and Burdick to study Riemannian metrics of\u0000positive Ricci curvature on connected sums, can appear as cross sections of\u0000tangent cones of the same point.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quasihyperbolic metric and Gromov hyperbolicity spaces 准双曲度量和格罗莫夫双曲空间
arXiv - MATH - Metric Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.12006
Hongjun Liu, Ling Xia, Shasha Yan
{"title":"Quasihyperbolic metric and Gromov hyperbolicity spaces","authors":"Hongjun Liu, Ling Xia, Shasha Yan","doi":"arxiv-2409.12006","DOIUrl":"https://doi.org/arxiv-2409.12006","url":null,"abstract":"In this paper, we introduce the concepts of short arc and length map in\u0000quasihyperbolic metric spaces, and obtain some geometric characterizations of\u0000Gromov hyperbolicity for quasihyperbolic metric spaces in terms of the\u0000properties of short arc and length map.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Tiling with Three Polygons is Undecidable 用三个多边形平铺是不可判定的
arXiv - MATH - Metric Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11582
Erik D. Demaine, Stefan Langerman
{"title":"Tiling with Three Polygons is Undecidable","authors":"Erik D. Demaine, Stefan Langerman","doi":"arxiv-2409.11582","DOIUrl":"https://doi.org/arxiv-2409.11582","url":null,"abstract":"We prove that the following problem is co-RE-complete and thus undecidable:\u0000given three simple polygons, is there a tiling of the plane where every tile is\u0000an isometry of one of the three polygons (either allowing or forbidding\u0000reflections)? This result improves on the best previous construction which\u0000requires five polygons.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Curvature-dimension condition of sub-Riemannian $α$-Grushin half-spaces 亚黎曼 $α$-Grushin 半空间的曲率维度条件
arXiv - MATH - Metric Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11177
Samuël Borza, Kenshiro Tashiro
{"title":"Curvature-dimension condition of sub-Riemannian $α$-Grushin half-spaces","authors":"Samuël Borza, Kenshiro Tashiro","doi":"arxiv-2409.11177","DOIUrl":"https://doi.org/arxiv-2409.11177","url":null,"abstract":"We provide new examples of sub-Riemannian manifolds with boundary equipped\u0000with a smooth measure that satisfy the $mathsf{RCD}(K , N)$ condition. They\u0000are constructed by equipping the half-plane, the hemisphere and the hyperbolic\u0000half-plane with a two-dimensional almost-Riemannian structure and a measure\u0000that vanishes on their boundary. The construction of these spaces is inspired\u0000from the geometry of the $alpha$-Grushin plane.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the classification of lattice polytopes via affine equivalence 通过仿射等价关系论格子多边形的分类
arXiv - MATH - Metric Geometry Pub Date : 2024-09-16 DOI: arxiv-2409.09985
Zhanyuan Cai, Yuqin Zhang, Qiuyue Liu
{"title":"On the classification of lattice polytopes via affine equivalence","authors":"Zhanyuan Cai, Yuqin Zhang, Qiuyue Liu","doi":"arxiv-2409.09985","DOIUrl":"https://doi.org/arxiv-2409.09985","url":null,"abstract":"In 1980, V. I. Arnold studied the classification problem for convex lattice\u0000polygons of a given area. Since then, this problem and its analogues have been\u0000studied by many authors, including B'ar'any, Lagarias, Pach, Santos, Ziegler\u0000and Zong. Despite extensive study, the structure of the representative sets in\u0000the classifications remains unclear, indicating a need for refined\u0000classification methods. In this paper, we propose a novel classification\u0000framework based on affine equivalence, which offers a fresh perspective on the\u0000problem. Our approach yields several classification results that extend and\u0000complement B'ar'any's work on volume and Zong's work on cardinality. These\u0000new results provide a more nuanced understanding of the structure of the\u0000representative set, offering deeper insights into the classification problem.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Deforming the weighted-homogeneous foliation, and trivializing families of semi-weighted homogeneous ICIS 加权均质叶状变形，以及半加权均质 ICIS 族的微观化
arXiv - MATH - Metric Geometry Pub Date : 2024-09-15 DOI: arxiv-2409.09764
Dmitry Kerner, Rodrigo Mendes
{"title":"Deforming the weighted-homogeneous foliation, and trivializing families of semi-weighted homogeneous ICIS","authors":"Dmitry Kerner, Rodrigo Mendes","doi":"arxiv-2409.09764","DOIUrl":"https://doi.org/arxiv-2409.09764","url":null,"abstract":"Let X_o be a weighted-homogeneous complete intersection germ in (R^N,o) or\u0000(C^N,o), with arbitrary singularities, possibly non-reduced. Take the foliation\u0000of the ambient space by weighted-homogeneous real arcs, ga_s. Take a deformation of X_o by higher order terms, X_t. Does the foliation\u0000ga_s deform compatibly with X_t? We identify the ``obstruction locus\", Sigma\u0000in X_o, outside of which such a deformation does exist, and possesses\u0000exceptionally nice properties. Using this deformed foliation we construct a contact trivialization of the\u0000family of defining equations by a homeomorphism that is real analytic (resp.\u0000Nash) off the origin, differentiable at the origin, whose presentation in\u0000weighted-polar coordinates is globally real-analytic (resp. globally Nash), and\u0000with controlled Lipschitz/C^1-properties.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Globally Rigid Convex Braced Polygons 全局刚性凸支撑多边形
arXiv - MATH - Metric Geometry Pub Date : 2024-09-14 DOI: arxiv-2409.09465
Robert Connelly, Bill Jackson, Shin-ichi Tanigawa, Zhen Zhang
{"title":"Globally Rigid Convex Braced Polygons","authors":"Robert Connelly, Bill Jackson, Shin-ichi Tanigawa, Zhen Zhang","doi":"arxiv-2409.09465","DOIUrl":"https://doi.org/arxiv-2409.09465","url":null,"abstract":"Here we propose a class of frameworks in the plane, braced polygons, that may\u0000be globally rigid and are analogous to convex polyopes in 3 space that are\u0000rigid by Cauchy's rigidity Theorem in 1813.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Magnetic Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on Riemannian manifolds 黎曼流形上的磁性布伦-闵科夫斯基不等式和博雷尔-布拉斯坎普-利布不等式
arXiv - MATH - Metric Geometry Pub Date : 2024-09-12 DOI: arxiv-2409.08001
Rotem Assouline
{"title":"Magnetic Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on Riemannian manifolds","authors":"Rotem Assouline","doi":"arxiv-2409.08001","DOIUrl":"https://doi.org/arxiv-2409.08001","url":null,"abstract":"We study the Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on a\u0000Riemannian manifold endowed with an exact magnetic field, replacing geodesics\u0000by minimizers of an action functional given by the length minus the integral of\u0000the magnetic potential. We prove that nonnegativity of a suitably defined\u0000magnetic Ricci curvature implies a magnetic Brunn-Minkowski inequality. More\u0000generally, given an arbitrary volume form on the manifold, we introduce a\u0000weighted magnetic Ricci curvature, and prove a magnetic version of the\u0000Borell-Brascamp-Lieb inequality.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A new upper bound for codes with a single Hamming distance 单一汉明距离编码的新上限
arXiv - MATH - Metric Geometry Pub Date : 2024-09-12 DOI: arxiv-2409.07877
Gábor Hegedüs
{"title":"A new upper bound for codes with a single Hamming distance","authors":"Gábor Hegedüs","doi":"arxiv-2409.07877","DOIUrl":"https://doi.org/arxiv-2409.07877","url":null,"abstract":"In this short note we give a new upper bound for the size of a set family\u0000with a single Hamming distance. Our proof is an application of the linear algebra bound method.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Variations on a theme of empty polytopes 空多边形主题变奏曲
arXiv - MATH - Metric Geometry Pub Date : 2024-09-11 DOI: arxiv-2409.07262
Srinivas Arun, Travis Dillon
{"title":"Variations on a theme of empty polytopes","authors":"Srinivas Arun, Travis Dillon","doi":"arxiv-2409.07262","DOIUrl":"https://doi.org/arxiv-2409.07262","url":null,"abstract":"Given a set $S subseteq mathbb{R}^d$, an empty polytope has vertices in $S$\u0000but contains no other point of $S$. Empty polytopes are closely related to\u0000so-called Helly numbers, which extend Helly's theorem to more general point\u0000sets in $mathbb{R}^d$. We improve bounds on the number of vertices in empty\u0000polytopes in exponential lattices, arithmetic congruence sets, and 2-syndetic\u0000sets. We also study hollow polytopes, which have vertices in $S$ and no points of\u0000$S$ in their interior. We obtain bounds on the number of vertices in hollow\u0000polytopes under certain conditions, such as the vertices being in general\u0000position. Finally, we obtain relatively tight asymptotic bounds for polytopes which do\u0000not contain lattice segments of large length.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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