By using optimal transport theory, we establish a sharp Alexandroff--Bakelman--Pucci (ABP) type estimate on metric measure spaces with synthetic Riemannian Ricci curvature lower bounds, and prove some geometric and functional inequalities including a functional ABP estimate. Our result not only extends the border of ABP estimate, but also provides an effective substitution of Jacobi fields computation in the non-smooth framework, which has potential applications to many problems in non-smooth geometric analysis.
{"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":"59 1","pages":""},"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}
The definition of Ricci curvature on graphs in Bakry-'Emery's sense based on curvature dimension condition was introduced by Lin and Yau [emph{Math. Res. Lett.}, 2010]. Hua and Lin [emph{Comm. Anal. Geom.}, 2019] classified unweighted graphs satisfying the curvature dimension condition $CD(0,infty)$ whose girth are at least five. In this paper, we classify all of connected unweighted normalized $C_4$-free graphs satisfying curvature dimension condition $CD(0,infty)$ for minimum degree at least 2 and the case with non-normalized Laplacian without degree condition..
Lin和Yau[emph{Math. Res.Lett.}, 2010]引入了基于曲率维条件的Bakry-'Emery意义上的图的里奇曲率定义。Hua 和 Lin [emph{Comm. Anal. Geom.}，2019] 对满足曲率维条件 $CD(0,infty)$、周长至少为五的无权重图进行了分类。本文分类了所有满足曲率维度条件$CD(0,infty)$的最小阶数至少为2的无连接无权重归一化$C_4$无图，以及无阶数条件的非归一化拉普拉斯图...
{"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":"30 1","pages":""},"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}
This paper is devoted to the study of the $m$-point homogeneity property for the vertex sets of polytopes in Euclidean spaces. In particular, we present the classifications of $2$-point and $3$-point homogeneous polyhedra in $mathbb{R}^3$.
{"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":"160 1","pages":""},"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}
Sublinearly Morse boundaries of proper geodesic spaces are introduced by Qing, Rafi and Tiozzo. Expanding on this work, Qing and Rafi recently developed the quasi-redirecting boundary, denoted $partial G$, to include all directions of metric spaces at infinity. Both boundaries are topological spaces that consist of equivalence classes of quasi-geodesic rays and are quasi-isometrically invariant. In this paper, we study these boundaries when the space is equipped with a geometric group action. In particular, we show that $G$ acts minimally on $partial_kappa G$ and that contracting elements of G induces a weak north-south dynamic on $partial_kappa G$. We also prove, when $partial G$ exists and $|partial_kappa G|geq3$, $G$ acts minimally on $partial G$ and $partial G$ is a second countable topological space. The last section concerns the restriction to proper CAT(0) spaces and finite dimensional CAT cube complexes. We show that when $G$ acts geometrically on a finite dimensional CAT(0) cube complex (whose QR boundary is assumed to exist), then a nontrivial QR boundary implies the existence of a Morse element in $G$. Lastly, we show that if $X$ is a proper cocompact CAT(0) space, then $partial G$ is a visibility space.
{"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":"32 1","pages":""},"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}
In this paper we study isometries of quantum Wasserstein distances and divergences on the quantum bit state space. We describe isometries with respect to the symmetric quantum Wasserstein divergence $d_{sym}$, the divergence induced by all of the Pauli matrices. We also give a complete characterization of isometries with respect to $D_z$, the quantum Wasserstein distance corresponding to the single Pauli matrix $sigma_z$.
{"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":"18 1","pages":""},"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}
A well-known theorem of Assouad states that metric spaces satisfying the doubling property can be snowflaked and bi-Lipschitz embedded into Euclidean spaces. Due to the invariance of many geometric properties under bi-Lipschitz maps, this result greatly facilitates the study of such spaces. We prove a non-injective analog of this embedding theorem for spaces of finite Minkowski dimension. This allows for non-doubling spaces to be weakly embedded and studied in the usual Euclidean setting. Such spaces often arise in the context of random geometry and mathematical physics with the Brownian continuum tree and Liouville quantum gravity metrics being prominent examples.
{"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":"27 1","pages":""},"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}
We use Lie sphere geometry to describe two large categories of generalized Voronoi diagrams that can be encoded in terms of the Lie quadric, the Lie inner product, and polyhedra. The first class consists of diagrams defined in terms of extremal spheres in the space of Lie spheres, and the second class includes minimization diagrams for functions that can be expressed in terms of affine functions on a higher-dimensional space. These results unify and generalize previous descriptions of generalized Voronoi diagrams as convex hull problems. Special cases include classical Voronoi diagrams, power diagrams, order $k$ and farthest point diagrams, Apollonius diagrams, medial axes, and generalized Voronoi diagrams whose sites are combinations of points, spheres and half-spaces. We describe the application of these results to algorithms for computing generalized Voronoi diagrams and find the complexity of these algorithms.
{"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":"8 1","pages":""},"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}
It is shown that if $A subseteq mathbb{R}^3$ is a Borel set of Hausdorff dimension $dim A>1$, and if $rho_{theta}$ is orthogonal projection to the line spanned by $left( cos theta, sin theta, 1 right)$, then $rho_{theta}(A)$ has positive length for all $theta$ outside a set of Hausdorff dimension $frac{3-dim A}{2}$.
{"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":"77 1","pages":""},"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}
The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in that the edge weights contain the cotangents of certain combinations of angles and are non-negative if and only if the triangulation is Delaunay. On the other hand, these discretizations are structure-preserving in several respects. We prove that the area of a convex polyhedron can be written in terms of the discrete spherical Laplacian of the support function, whose expression is the same as the area of a smooth convex body in terms of the usual spherical Laplacian. We show that the conformal factors of discrete conformal vector fields on a triangulated surface of curvature $k in {-1,1}$ are $-2k$-eigenfunctions of our discrete Laplacians, exactly as in the smooth setting. The discrete conformality can be understood here both in the sense of the vertex scaling and in the sense of circle patterns. Finally, we connect the $-2k$-eigenfunctions to infinitesimal isometric deformations of a polyhedron inscribed into corresponding quadrics.
欧几里得三角曲面上的离散拉普拉斯是一个早已确立的概念。我们介绍球面和双曲三角面上的离散拉普拉斯。一方面，我们的定义与欧几里得的定义很接近，即边权重包含某些角组合的余切，并且只有当三角剖分是德劳内时，边权重才是非负的。另一方面，这些离散化在很多方面都是结构保留的。我们证明了凸多面体的面积可以用支撑函数的离散球面拉普拉奇来表示，其表达式与用实际球面拉普拉奇表示的光滑凸体的面积相同。我们证明，在曲率为 $k in {-1,1}$ 的三角曲面上，离散共形向量场的共形因子是离散拉普拉斯的 $-2k$ 特征函数，这与平滑设置中的情况完全相同。这里的离散保角既可以从顶点缩放的意义上理解，也可以从圆模式的意义上理解。最后，我们将$-2k$特征函数与刻入相应四边形的多面体的无限小等距变形联系起来。
{"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":"4 1","pages":""},"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}
MIT CompGeom Group, Hugo A. Akitaya, Erik D. Demaine, Adam Hesterberg, Anna Lubiw, Jayson Lynch, Joseph O'Rourke, Frederick Stock, Josef Tkadlec
A polyiamond is a polygon composed of unit equilateral triangles, and a generalized deltahedron is a convex polyhedron whose every face is a convex polyiamond. We study a variant where one face may be an exception. For a convex polygon P, if there is a convex polyhedron that has P as one face and all the other faces are convex polyiamonds, then we say that P can be domed. Our main result is a complete characterization of which equiangular n-gons can be domed: only if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the integer edge lengths.
多面体是由单位等边三角形组成的多边形，广义正三角形是每个面都是凸多面体的凸多面体。我们研究的是其中一个面可能是例外的变体。对于凸多边形 P，如果有一个凸多面体以 P 为一个面，而其他所有面都是凸多面体，那么我们就说 P 可以是圆顶的。我们的主要结果是完整地描述了哪些等角 n 边形可以被穹顶化：只有当 n 在 {3, 4, 5, 6, 8, 10, 12} 中，并且只有在边长为整数的某些条件下，这些等角 n 边形才可以被穹顶化。
{"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":"57 1","pages":""},"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}
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