We give new examples of manifolds that appear as cross sections of tangent cones of non-collapsed Ricci limit spaces. It was shown by Colding-Naber that the homeomorphism types of the tangent cones of a fixed point of such a space do not need to be unique. In fact, they constructed an example in dimension 5 where two different homeomorphism types appear at the same point. In this note, we extend this result and construct limit spaces in all dimensions at least 5 where any finite collection of manifolds that admit core metrics, a type of metric introduced by Perelman and Burdick to study Riemannian metrics of positive Ricci curvature on connected sums, can appear as cross sections of tangent cones of the same point.
{"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}
In this paper, we introduce the concepts of short arc and length map in quasihyperbolic metric spaces, and obtain some geometric characterizations of Gromov hyperbolicity for quasihyperbolic metric spaces in terms of the properties of short arc and length map.
{"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}
We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding reflections)? This result improves on the best previous construction which requires five polygons.
{"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}
We provide new examples of sub-Riemannian manifolds with boundary equipped with a smooth measure that satisfy the $mathsf{RCD}(K , N)$ condition. They are constructed by equipping the half-plane, the hemisphere and the hyperbolic half-plane with a two-dimensional almost-Riemannian structure and a measure that vanishes on their boundary. The construction of these spaces is inspired from the geometry of the $alpha$-Grushin plane.
{"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}
In 1980, V. I. Arnold studied the classification problem for convex lattice polygons of a given area. Since then, this problem and its analogues have been studied by many authors, including B'ar'any, Lagarias, Pach, Santos, Ziegler and Zong. Despite extensive study, the structure of the representative sets in the classifications remains unclear, indicating a need for refined classification methods. In this paper, we propose a novel classification framework based on affine equivalence, which offers a fresh perspective on the problem. Our approach yields several classification results that extend and complement B'ar'any's work on volume and Zong's work on cardinality. These new results provide a more nuanced understanding of the structure of the representative set, offering deeper insights into the classification problem.
1980 年，V. I. Arnold 研究了给定面积的凸网格多边形的分类问题。此后，许多学者对这一问题及其类似问题进行了研究，包括 B'ar'any, Lagarias, Pach, Santos, Zieglerand Zong。尽管进行了广泛的研究，但分类中代表集的结构仍不清楚，这表明需要改进分类方法。在本文中，我们提出了一种基于仿射等价性的新型分类框架，为这一问题提供了全新的视角。我们的方法产生了几个分类结果，扩展并补充了 B'ar'any 在体积方面的工作和 Zong 在卡方性方面的工作。这些新结果提供了对表征集结构更细致入微的理解，为分类问题提供了更深刻的见解。
{"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}
Let X_o be a weighted-homogeneous complete intersection germ in (R^N,o) or (C^N,o), with arbitrary singularities, possibly non-reduced. Take the foliation of 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 ga_s deform compatibly with X_t? We identify the ``obstruction locus", Sigma in X_o, outside of which such a deformation does exist, and possesses exceptionally nice properties. Using this deformed foliation we construct a contact trivialization of the family of defining equations by a homeomorphism that is real analytic (resp. Nash) off the origin, differentiable at the origin, whose presentation in weighted-polar coordinates is globally real-analytic (resp. globally Nash), and with controlled Lipschitz/C^1-properties.
{"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}
Robert Connelly, Bill Jackson, Shin-ichi Tanigawa, Zhen Zhang
Here we propose a class of frameworks in the plane, braced polygons, that may be globally rigid and are analogous to convex polyopes in 3 space that are rigid by Cauchy's rigidity Theorem in 1813.
{"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}
We study the Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on a Riemannian manifold endowed with an exact magnetic field, replacing geodesics by minimizers of an action functional given by the length minus the integral of the magnetic potential. We prove that nonnegativity of a suitably defined magnetic Ricci curvature implies a magnetic Brunn-Minkowski inequality. More generally, given an arbitrary volume form on the manifold, we introduce a weighted magnetic Ricci curvature, and prove a magnetic version of the Borell-Brascamp-Lieb inequality.
{"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}
In this short note we give a new upper bound for the size of a set family with a single Hamming distance. Our proof is an application of the linear algebra bound method.
{"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}
Given a set $S subseteq mathbb{R}^d$, an empty polytope has vertices in $S$ but contains no other point of $S$. Empty polytopes are closely related to so-called Helly numbers, which extend Helly's theorem to more general point sets in $mathbb{R}^d$. We improve bounds on the number of vertices in empty polytopes in exponential lattices, arithmetic congruence sets, and 2-syndetic sets. We also study hollow polytopes, which have vertices in $S$ and no points of $S$ in their interior. We obtain bounds on the number of vertices in hollow polytopes under certain conditions, such as the vertices being in general position. Finally, we obtain relatively tight asymptotic bounds for polytopes which do not contain lattice segments of large length.
{"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}
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