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ABP estimate on metric measure spaces via optimal transport 通过最优传输对度量空间进行 ABP 估算
arXiv - MATH - Metric Geometry Pub Date : 2024-08-20 DOI: arxiv-2408.10725
Bang-Xian Han
{"title":"ABP estimate on metric measure spaces via optimal transport","authors":"Bang-Xian Han","doi":"arxiv-2408.10725","DOIUrl":"https://doi.org/arxiv-2408.10725","url":null,"abstract":"By using optimal transport theory, we establish a sharp\u0000Alexandroff--Bakelman--Pucci (ABP) type estimate on metric measure spaces with\u0000synthetic Riemannian Ricci curvature lower bounds, and prove some geometric and\u0000functional inequalities including a functional ABP estimate. Our result not\u0000only extends the border of ABP estimate, but also provides an effective\u0000substitution of Jacobi fields computation in the non-smooth framework, which\u0000has potential applications to many problems in non-smooth geometric analysis.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187643","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
Graphs with nonnegative Bakry-Émery curvature without Quadrilateral 无四边形的非负 Bakry-Émery 曲率图形
arXiv - MATH - Metric Geometry Pub Date : 2024-08-19 DOI: arxiv-2408.09823
Huiqiu Lin, Zhe You
{"title":"Graphs with nonnegative Bakry-Émery curvature without Quadrilateral","authors":"Huiqiu Lin, Zhe You","doi":"arxiv-2408.09823","DOIUrl":"https://doi.org/arxiv-2408.09823","url":null,"abstract":"The definition of Ricci curvature on graphs in Bakry-'Emery's sense based on\u0000curvature dimension condition was introduced by Lin and Yau [emph{Math. Res.\u0000Lett.}, 2010]. Hua and Lin [emph{Comm. Anal. Geom.}, 2019] classified\u0000unweighted graphs satisfying the curvature dimension condition $CD(0,infty)$\u0000whose girth are at least five. In this paper, we classify all of connected\u0000unweighted normalized $C_4$-free graphs satisfying curvature dimension\u0000condition $CD(0,infty)$ for minimum degree at least 2 and the case with\u0000non-normalized Laplacian without degree condition..","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187650","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 $m$-point homogeneous polyhedra in $3$-dimensional Euclidean space 论 3 美元欧几里得三维空间中的 m 美元点均质多面体
arXiv - MATH - Metric Geometry Pub Date : 2024-08-19 DOI: arxiv-2408.09911
V. N. Berestovskii, Yu. G. Nikonorov
{"title":"On $m$-point homogeneous polyhedra in $3$-dimensional Euclidean space","authors":"V. N. Berestovskii, Yu. G. Nikonorov","doi":"arxiv-2408.09911","DOIUrl":"https://doi.org/arxiv-2408.09911","url":null,"abstract":"This paper is devoted to the study of the $m$-point homogeneity property for\u0000the vertex sets of polytopes in Euclidean spaces. In particular, we present the\u0000classifications of $2$-point and $3$-point homogeneous polyhedra in\u0000$mathbb{R}^3$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187647","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
Topological and Dynamic Properties of the Sublinearly Morse Boundary and the Quasi-Redirecting Boundary 亚线性莫尔斯边界和准重定向边界的拓扑和动态特性
arXiv - MATH - Metric Geometry Pub Date : 2024-08-19 DOI: arxiv-2408.10105
Jacob Garcia, Yulan Qing, Elliott Vest
{"title":"Topological and Dynamic Properties of the Sublinearly Morse Boundary and the Quasi-Redirecting Boundary","authors":"Jacob Garcia, Yulan Qing, Elliott Vest","doi":"arxiv-2408.10105","DOIUrl":"https://doi.org/arxiv-2408.10105","url":null,"abstract":"Sublinearly Morse boundaries of proper geodesic spaces are introduced by\u0000Qing, Rafi and Tiozzo. Expanding on this work, Qing and Rafi recently developed\u0000the quasi-redirecting boundary, denoted $partial G$, to include all directions\u0000of metric spaces at infinity. Both boundaries are topological spaces that\u0000consist of equivalence classes of quasi-geodesic rays and are\u0000quasi-isometrically invariant. In this paper, we study these boundaries when\u0000the space is equipped with a geometric group action. In particular, we show\u0000that $G$ acts minimally on $partial_kappa G$ and that contracting elements of\u0000G induces a weak north-south dynamic on $partial_kappa G$. We also prove,\u0000when $partial G$ exists and $|partial_kappa G|geq3$, $G$ acts minimally on\u0000$partial G$ and $partial G$ is a second countable topological space. The last\u0000section concerns the restriction to proper CAT(0) spaces and finite dimensional\u0000CAT cube complexes. We show that when $G$ acts geometrically on a finite\u0000dimensional CAT(0) cube complex (whose QR boundary is assumed to exist), then a\u0000nontrivial QR boundary implies the existence of a Morse element in $G$. Lastly,\u0000we show that if $X$ is a proper cocompact CAT(0) space, then $partial G$ is a\u0000visibility space.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187645","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
Isometries of the qubit state space with respect to quantum Wasserstein distances 量子瓦瑟尔斯坦距离的量子比特状态空间等位性
arXiv - MATH - Metric Geometry Pub Date : 2024-08-19 DOI: arxiv-2408.09879
Richárd Simon, Dániel Virosztek
{"title":"Isometries of the qubit state space with respect to quantum Wasserstein distances","authors":"Richárd Simon, Dániel Virosztek","doi":"arxiv-2408.09879","DOIUrl":"https://doi.org/arxiv-2408.09879","url":null,"abstract":"In this paper we study isometries of quantum Wasserstein distances and\u0000divergences on the quantum bit state space. We describe isometries with respect\u0000to the symmetric quantum Wasserstein divergence $d_{sym}$, the divergence\u0000induced by all of the Pauli matrices. We also give a complete characterization\u0000of isometries with respect to $D_z$, the quantum Wasserstein distance\u0000corresponding to the single Pauli matrix $sigma_z$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187649","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
Minkowski weak embedding theorem 闵科夫斯基弱嵌入定理
arXiv - MATH - Metric Geometry Pub Date : 2024-08-17 DOI: arxiv-2408.09063
Efstathios Konstantinos Chrontsios Garitsis, Sascha Troscheit
{"title":"Minkowski weak embedding theorem","authors":"Efstathios Konstantinos Chrontsios Garitsis, Sascha Troscheit","doi":"arxiv-2408.09063","DOIUrl":"https://doi.org/arxiv-2408.09063","url":null,"abstract":"A well-known theorem of Assouad states that metric spaces satisfying the\u0000doubling property can be snowflaked and bi-Lipschitz embedded into Euclidean\u0000spaces. Due to the invariance of many geometric properties under bi-Lipschitz\u0000maps, this result greatly facilitates the study of such spaces. We prove a\u0000non-injective analog of this embedding theorem for spaces of finite Minkowski\u0000dimension. This allows for non-doubling spaces to be weakly embedded and\u0000studied in the usual Euclidean setting. Such spaces often arise in the context\u0000of random geometry and mathematical physics with the Brownian continuum tree\u0000and Liouville quantum gravity metrics being prominent examples.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187648","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
Generalized Voronoi Diagrams and Lie Sphere Geometry 广义沃罗诺图和烈球几何
arXiv - MATH - Metric Geometry Pub Date : 2024-08-17 DOI: arxiv-2408.09279
John Edwards, Tracy Payne, Elena Schafer
{"title":"Generalized Voronoi Diagrams and Lie Sphere Geometry","authors":"John Edwards, Tracy Payne, Elena Schafer","doi":"arxiv-2408.09279","DOIUrl":"https://doi.org/arxiv-2408.09279","url":null,"abstract":"We use Lie sphere geometry to describe two large categories of generalized\u0000Voronoi diagrams that can be encoded in terms of the Lie quadric, the Lie inner\u0000product, and polyhedra. The first class consists of diagrams defined in terms\u0000of extremal spheres in the space of Lie spheres, and the second class includes\u0000minimization diagrams for functions that can be expressed in terms of affine\u0000functions on a higher-dimensional space. These results unify and generalize\u0000previous descriptions of generalized Voronoi diagrams as convex hull problems.\u0000Special cases include classical Voronoi diagrams, power diagrams, order $k$ and\u0000farthest point diagrams, Apollonius diagrams, medial axes, and generalized\u0000Voronoi diagrams whose sites are combinations of points, spheres and\u0000half-spaces. We describe the application of these results to algorithms for\u0000computing generalized Voronoi diagrams and find the complexity of these\u0000algorithms.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187646","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
Exceptional sets for length under restricted families of projections onto lines in $mathbb{R}^3$ 在$mathbb{R}^3$中投影到直线的受限族下的长度异常集
arXiv - MATH - Metric Geometry Pub Date : 2024-08-09 DOI: arxiv-2408.04885
Terence L. J. Harris
{"title":"Exceptional sets for length under restricted families of projections onto lines in $mathbb{R}^3$","authors":"Terence L. J. Harris","doi":"arxiv-2408.04885","DOIUrl":"https://doi.org/arxiv-2408.04885","url":null,"abstract":"It is shown that if $A subseteq mathbb{R}^3$ is a Borel set of Hausdorff\u0000dimension $dim A>1$, and if $rho_{theta}$ is orthogonal projection to the\u0000line spanned by $left( cos theta, sin theta, 1 right)$, then\u0000$rho_{theta}(A)$ has positive length for all $theta$ outside a set of\u0000Hausdorff dimension $frac{3-dim A}{2}$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934874","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
Discrete Laplacians -- spherical and hyperbolic 离散拉普拉斯 -- 球面和双曲
arXiv - MATH - Metric Geometry Pub Date : 2024-08-09 DOI: arxiv-2408.04877
Ivan Izmestiev, Wai Yeung Lam
{"title":"Discrete Laplacians -- spherical and hyperbolic","authors":"Ivan Izmestiev, Wai Yeung Lam","doi":"arxiv-2408.04877","DOIUrl":"https://doi.org/arxiv-2408.04877","url":null,"abstract":"The discrete Laplacian on Euclidean triangulated surfaces is a\u0000well-established notion. We introduce discrete Laplacians on spherical and\u0000hyperbolic triangulated surfaces. On the one hand, our definitions are close to\u0000the Euclidean one in that the edge weights contain the cotangents of certain\u0000combinations of angles and are non-negative if and only if the triangulation is\u0000Delaunay. On the other hand, these discretizations are structure-preserving in\u0000several respects. We prove that the area of a convex polyhedron can be written\u0000in terms of the discrete spherical Laplacian of the support function, whose\u0000expression is the same as the area of a smooth convex body in terms of the\u0000usual spherical Laplacian. We show that the conformal factors of discrete\u0000conformal vector fields on a triangulated surface of curvature $k in {-1,1}$\u0000are $-2k$-eigenfunctions of our discrete Laplacians, exactly as in the smooth\u0000setting. The discrete conformality can be understood here both in the sense of\u0000the vertex scaling and in the sense of circle patterns. Finally, we connect the\u0000$-2k$-eigenfunctions to infinitesimal isometric deformations of a polyhedron\u0000inscribed into corresponding quadrics.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934618","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
Deltahedral Domes over Equiangular Polygons 等边多边形上的三角穹顶
arXiv - MATH - Metric Geometry Pub Date : 2024-08-08 DOI: arxiv-2408.04687
MIT CompGeom Group, Hugo A. Akitaya, Erik D. Demaine, Adam Hesterberg, Anna Lubiw, Jayson Lynch, Joseph O'Rourke, Frederick Stock, Josef Tkadlec
{"title":"Deltahedral Domes over Equiangular Polygons","authors":"MIT CompGeom Group, Hugo A. Akitaya, Erik D. Demaine, Adam Hesterberg, Anna Lubiw, Jayson Lynch, Joseph O'Rourke, Frederick Stock, Josef Tkadlec","doi":"arxiv-2408.04687","DOIUrl":"https://doi.org/arxiv-2408.04687","url":null,"abstract":"A polyiamond is a polygon composed of unit equilateral triangles, and a\u0000generalized deltahedron is a convex polyhedron whose every face is a convex\u0000polyiamond. We study a variant where one face may be an exception. For a convex\u0000polygon P, if there is a convex polyhedron that has P as one face and all the\u0000other faces are convex polyiamonds, then we say that P can be domed. Our main\u0000result is a complete characterization of which equiangular n-gons can be domed:\u0000only if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the\u0000integer edge lengths.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934619","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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