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Lipschitz extensions from spaces of nonnegative curvature into CAT(1) spaces 从非负曲率空间到 CAT(1) 空间的 Lipschitz 扩展
arXiv - MATH - Metric Geometry Pub Date : 2024-08-01 DOI: arxiv-2408.00564
Sebastian Gietl
{"title":"Lipschitz extensions from spaces of nonnegative curvature into CAT(1) spaces","authors":"Sebastian Gietl","doi":"arxiv-2408.00564","DOIUrl":"https://doi.org/arxiv-2408.00564","url":null,"abstract":"We prove that complete $text{CAT}(kappa)$ spaces of sufficiently small\u0000radii possess metric cotype 2 and metric Markov cotype 2. This generalizes the\u0000previously known result for complete $text{CAT}(0)$ spaces. The generalization\u0000involves extending the variance inequality known for barycenters in\u0000$text{CAT}(0)$ spaces to an inequality analogous to one for 2-uniformly convex\u0000Banach spaces, and demonstrating that the barycenter map on such spaces is\u0000Lipschitz continuous on the corresponding Wasserstein 2 space. By utilizing the\u0000generalized Ball extension theorem by Mendel and Naor, we obtain an extension\u0000result for Lipschitz maps from Alexandrov spaces of nonnegative curvature into\u0000$text{CAT}(kappa)$ spaces whose image is contained in a subspace of\u0000sufficiently small radius, thereby weakening the curvature assumption in the\u0000well-known Lipschitz extension theorem for Alexandrov spaces by Lang and\u0000Schr\"oder. As an additional application, we obtain that $ell_p$ spaces for $p\u0000> 2$ cannot be uniformly embedded into any $text{CAT}(kappa)$ space with\u0000sufficiently small diameter.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882944","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
Comments on the regularity of harmonic maps between singular spaces 关于奇异空间之间谐波映射规律性的评论
arXiv - MATH - Metric Geometry Pub Date : 2024-08-01 DOI: arxiv-2408.00479
Luca Gennaioli, Nicola Gigli, Hui-Chun Zhang, Xi-Ping Zhu
{"title":"Comments on the regularity of harmonic maps between singular spaces","authors":"Luca Gennaioli, Nicola Gigli, Hui-Chun Zhang, Xi-Ping Zhu","doi":"arxiv-2408.00479","DOIUrl":"https://doi.org/arxiv-2408.00479","url":null,"abstract":"In this work we are going to establish H\"older continuity of harmonic maps\u0000from an open set $Omega$ in an ${rm RCD}(K,N)$ space valued into a ${rm\u0000CAT}(kappa)$ space, with the constraint that the image of $Omega$ via the map\u0000is contained in a sufficiently small ball in the target. Building on top of\u0000this regularity and assuming a local Lipschitz regularity of the map, we\u0000establish a weak version of the Bochner-Eells-Sampson inequality in such a\u0000non-smooth setting. Finally we study the boundary regularity of such maps.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882942","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
New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in $mathbf{H}^2!times!mathbf{R}$ space $mathbf{H}^2!
arXiv - MATH - Metric Geometry Pub Date : 2024-07-31 DOI: arxiv-2407.21251
Arnasli Yahya, Jenő Szirmai
{"title":"New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in $mathbf{H}^2!times!mathbf{R}$ space","authors":"Arnasli Yahya, Jenő Szirmai","doi":"arxiv-2407.21251","DOIUrl":"https://doi.org/arxiv-2407.21251","url":null,"abstract":"In this paper, we present a new record for the densest geodesic congruent\u0000ball packing configurations in $mathbf{H}^2!times!mathbf{R}$ geometry,\u0000generated by screw motion groups. These groups are derived from the direct\u0000product of rotational groups on $mathbf{H}^2$ and some translation components\u0000on the real fibre direction $mathbf{R}$ that can be determined by the\u0000corresponding Frobenius congruences. Moreover, we developed a procedure to\u0000determine the optimal radius for the densest geodesic ball packing\u0000configurations related to the considered screw motion groups. The highest\u0000packing density, $approx0.80529$, is achieved by a multi-transitive case given\u0000by rotational parameters $(2,20,4)$. E. Moln'{a}r demonstrated that\u0000homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere\u0000$mathcal{PS}^3(mathbf{V}^4, boldsymbol{V}_4, mathbf{R})$. We use this\u0000projective model of $mathbf{H}^2!times!mathbf{R}$ to compute and visualize\u0000the locally optimal geodesic ball arrangements.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870047","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
The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction 奥恩斯坦-乌伦贝克算子第一特征值的布伦-闵科夫斯基不等式和相关特征函数的对数凹性
arXiv - MATH - Metric Geometry Pub Date : 2024-07-31 DOI: arxiv-2407.21354
Andrea Colesanti, Elisa Francini, Galyna Livshyts, Paolo Salani
{"title":"The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction","authors":"Andrea Colesanti, Elisa Francini, Galyna Livshyts, Paolo Salani","doi":"arxiv-2407.21354","DOIUrl":"https://doi.org/arxiv-2407.21354","url":null,"abstract":"We prove that the first (nontrivial) Dirichlet eigenvalue of the\u0000Ornstein-Uhlenbeck operator $$ L(u)=Delta u-langlenabla u,xrangle,, $$ as\u0000a function of the domain, is convex with respect to the Minkowski addition, and\u0000we characterize the equality cases in some classes of convex sets. We also\u0000prove that the corresponding (positive) eigenfunction is log-concave if the\u0000domain is convex.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870048","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
Vertex generated polytopes 顶点生成的多边形
arXiv - MATH - Metric Geometry Pub Date : 2024-07-30 DOI: arxiv-2407.20604
Shiri Artstein-Avidan, Tomer Falah, Boaz A. Slomka
{"title":"Vertex generated polytopes","authors":"Shiri Artstein-Avidan, Tomer Falah, Boaz A. Slomka","doi":"arxiv-2407.20604","DOIUrl":"https://doi.org/arxiv-2407.20604","url":null,"abstract":"In this paper we define and investigate a class of polytopes which we call\u0000\"vertex generated\" consisting of polytopes which are the average of their $0$\u0000and $n$ dimensional faces. We show many results regarding this class, among\u0000them: that the class contains all zonotopes, that it is dense in dimension\u0000$n=2$, that any polytope can be summed with a zonotope so that the sum is in\u0000this class, and that a strong form of the celebrated \"Maurey Lemma\" holds for\u0000polytopes in this class. We introduce for every polytope a parameter which\u0000measures how far it is from being vertex-generated, and show that when this\u0000parameter is small, strong covering properties hold.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870049","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
Colorful positive bases decomposition and Helly-type results for cones 锥体的彩色正基分解和赫利型结果
arXiv - MATH - Metric Geometry Pub Date : 2024-07-30 DOI: arxiv-2407.20961
Grigory Ivanov
{"title":"Colorful positive bases decomposition and Helly-type results for cones","authors":"Grigory Ivanov","doi":"arxiv-2407.20961","DOIUrl":"https://doi.org/arxiv-2407.20961","url":null,"abstract":"We prove the following colorful Helly-type result: Fix $k in [d-1]$. Assume\u0000$mathcal{A}_1, dots, mathcal{A}_{d+(d-k)+1}$ are finite sets (colors) of\u0000nonzero vectors in $R^d$. If for every rainbow sub-selection $R$ from these\u0000sets of size at most $max {d+1, 2(d-k+1)}$, the system $langle {a},{x}\u0000rangle leq 0,; a in R$ has at least $k$ linearly independent solutions,\u0000then at least one of the systems $langle {a},{x} rangle leq 0,; a in\u0000mathcal{A}_i,$ $i in [d+(d-k)+1]$ has at least $k$ linearly independent\u0000solutions. A emph{rainbow sub-selection} from several sets refers to choosing at most\u0000one element from each set (color). The Helly-number $max {d+1, 2(d-k+1)}$ and the number of colors\u0000$d+(d-k)+1$ are optimal. Our key observation is a certain colorful Carath'eodory-type result for\u0000positive bases.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870119","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
Hölder regularity of harmonic functions on metric measure spaces 公度量空间上谐函数的荷尔德正则性
arXiv - MATH - Metric Geometry Pub Date : 2024-07-30 DOI: arxiv-2407.20789
Jin Gao, Meng Yang
{"title":"Hölder regularity of harmonic functions on metric measure spaces","authors":"Jin Gao, Meng Yang","doi":"arxiv-2407.20789","DOIUrl":"https://doi.org/arxiv-2407.20789","url":null,"abstract":"We introduce the H\"older regularity condition for harmonic functions on\u0000metric measure spaces and prove that under mild volume regular condition and\u0000upper heat kernel estimate, the H\"older regularity condition, the weak\u0000Bakry-'Emery non-negative curvature condition, the heat kernel H\"older\u0000continuity with or without exponential terms and the heat kernel near-diagonal\u0000lower bound are equivalent. As applications, firstly, we prove the validity of\u0000the so-called generalized reverse H\"older inequality on the Sierpi'nski\u0000carpet cable system, which was left open by Devyver, Russ, Yang (Int. Math.\u0000Res. Not. IMRN (2023), no. 18, 15537-15583). Secondly, we prove that two-sided\u0000heat kernel estimates alone imply gradient estimate for the heat kernel on\u0000strongly recurrent fractal-like cable systems, which improves the main results\u0000of the aforementioned paper. Thirdly, we obtain H\"older (Lipschitz) estimate\u0000for heat kernel on general metric measure spaces, which extends the classical\u0000Li-Yau gradient estimate for heat kernel on Riemannian manifolds.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870120","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 separability in discrete geometry 论离散几何中的可分离性
arXiv - MATH - Metric Geometry Pub Date : 2024-07-29 DOI: arxiv-2407.20169
Károly Bezdek, Zsolt Lángi
{"title":"On separability in discrete geometry","authors":"Károly Bezdek, Zsolt Lángi","doi":"arxiv-2407.20169","DOIUrl":"https://doi.org/arxiv-2407.20169","url":null,"abstract":"A problem of ErdH{o}s (Amer. Math. Monthly 52: 494-498, 1945) and a theorem\u0000of Fejes T'oth and Fejes T'oth (Acta Math. Acad. Sci. Hungar. 24: 229-232,\u00001973) initiated the study of non-separable arrangements of convex bodies and\u0000the investigation of totally separable packings of convex bodies with both\u0000topics analyzing the concept of separability from the point view of discrete\u0000geometry. This article surveys the progress made on these and some closely\u0000related problems and highlights the relevant questions that have been left\u0000open.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870122","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
Embedding Borel graphs into grids of asymptotically optimal dimension 将 Borel 图嵌入近似最优维度的网格中
arXiv - MATH - Metric Geometry Pub Date : 2024-07-29 DOI: arxiv-2407.19785
Anton Bernshteyn, Jing Yu
{"title":"Embedding Borel graphs into grids of asymptotically optimal dimension","authors":"Anton Bernshteyn, Jing Yu","doi":"arxiv-2407.19785","DOIUrl":"https://doi.org/arxiv-2407.19785","url":null,"abstract":"Let $G$ be a Borel graph all of whose finite subgraphs embed into the\u0000$d$-dimensional grid with diagonals. We show that then $G$ itself admits a\u0000Borel embedding into the Schreier graph of a free Borel action of $mathbb\u0000Z^{O(d)}$. This strengthens an earlier result of the authors, in which $O(d)$\u0000is replaced by $O(rho log rho)$, where $rho$ is the polynomial growth rate\u0000of $G$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870125","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
The Complement of a Closed Set Satisfying the Extended Exterior Sphere Condition 满足扩展外球面条件的封闭集的补集
arXiv - MATH - Metric Geometry Pub Date : 2024-07-29 DOI: arxiv-2407.20376
Chadi Nour, Jean Takche
{"title":"The Complement of a Closed Set Satisfying the Extended Exterior Sphere Condition","authors":"Chadi Nour, Jean Takche","doi":"arxiv-2407.20376","DOIUrl":"https://doi.org/arxiv-2407.20376","url":null,"abstract":"We provide a novel analytical proof of an improved version of [10, Theorem\u00003.1], showing that the complement of a closed set satisfying the extended\u0000exterior sphere condition is nothing but the union of closed balls with lower\u0000semicontinuous radius function. The improvement lies in the radius function,\u0000which is now larger than the one used in [10, Theorem 3.1].","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870050","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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