Abstract We show that if there is a smooth function f on a Finsler n-space M satisfying Δ2f = −kfgΔf for a positive constant k, then M is diffeomorphic with the n-sphere 𝕊n, where g denotes the weighted Riemannian metric. Moreover, we further show that the manifold is isometric to a Finsler sphere if the Ricci curvature is bounded below by (n − [one.tf])k and the S-curvature vanishes.
{"title":"Certain Conditions for a Finsler Manifold to Be Isometric with a Finsler Sphere","authors":"S. Yin, Huarong Wang","doi":"10.1515/agms-2022-0142","DOIUrl":"https://doi.org/10.1515/agms-2022-0142","url":null,"abstract":"Abstract We show that if there is a smooth function f on a Finsler n-space M satisfying Δ2f = −kfgΔf for a positive constant k, then M is diffeomorphic with the n-sphere 𝕊n, where g denotes the weighted Riemannian metric. Moreover, we further show that the manifold is isometric to a Finsler sphere if the Ricci curvature is bounded below by (n − [one.tf])k and the S-curvature vanishes.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"290 - 296"},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41547742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré inequality yet have empty interior. These examples are formed from an iterative process that involves removing well-behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes families of curves, we include a new way of obtaining Poincaré inequalities through the use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we further introduce the notion of of isoperimetric inequalities at given density levels and a way to iterate such inequalities. These tools are presented and apply to general metric measure measures. Our examples subsume the previous results of Mackay, Tyson, and Wildrick regarding non-self similar Sierpiński carpets, and extend them to many more general shapes as well as higher dimensions.
{"title":"Isoperimetric and Poincaré Inequalities on Non-Self-Similar Sierpiński Sponges: the Borderline Case","authors":"S. Eriksson-Bique, Jasun Gong","doi":"10.1515/agms-2022-0144","DOIUrl":"https://doi.org/10.1515/agms-2022-0144","url":null,"abstract":"Abstract In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré inequality yet have empty interior. These examples are formed from an iterative process that involves removing well-behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes families of curves, we include a new way of obtaining Poincaré inequalities through the use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we further introduce the notion of of isoperimetric inequalities at given density levels and a way to iterate such inequalities. These tools are presented and apply to general metric measure measures. Our examples subsume the previous results of Mackay, Tyson, and Wildrick regarding non-self similar Sierpiński carpets, and extend them to many more general shapes as well as higher dimensions.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"373 - 393"},"PeriodicalIF":1.0,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46620796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov–Hausdorff distance.We then construct branching geodesics of the Gromov–Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov– Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.
{"title":"Branching Geodesics of the Gromov-Hausdorff Distance","authors":"Yoshito Ishiki","doi":"10.1515/agms-2022-0136","DOIUrl":"https://doi.org/10.1515/agms-2022-0136","url":null,"abstract":"Abstract In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov–Hausdorff distance.We then construct branching geodesics of the Gromov–Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov– Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"109 - 128"},"PeriodicalIF":1.0,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45022196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A charge space (X, 𝒜, µ) is a generalisation of a measure space, consisting of a sample space X, a field of subsets 𝒜 and a finitely additive measure µ, also known as a charge. Properties a real-valued function on X may possess include T1-measurability and integrability. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of T1-measurability and integrability for a bounded charge space (µ(X) < ∞). These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that T1-measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if (X, 𝒜, µ) is a complete measure space. New characterisations of equality almost everywhere of two real-valued functions on a bounded charge space are provided. Necessary and sufficient conditions for the function space L1(X, 𝒜, µ) to be a Banach space are determined. Lastly, the concept of completion of a measure space is generalised for charge spaces, and it is shown that under certain conditions, completion of a charge space adds no new equivalence classes to the quotient space ℒp(X, 𝒜, µ).
{"title":"Properties of Functions on a Bounded Charge Space","authors":"J. Keith","doi":"10.1515/agms-2022-0134","DOIUrl":"https://doi.org/10.1515/agms-2022-0134","url":null,"abstract":"Abstract A charge space (X, 𝒜, µ) is a generalisation of a measure space, consisting of a sample space X, a field of subsets 𝒜 and a finitely additive measure µ, also known as a charge. Properties a real-valued function on X may possess include T1-measurability and integrability. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of T1-measurability and integrability for a bounded charge space (µ(X) < ∞). These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that T1-measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if (X, 𝒜, µ) is a complete measure space. New characterisations of equality almost everywhere of two real-valued functions on a bounded charge space are provided. Necessary and sufficient conditions for the function space L1(X, 𝒜, µ) to be a Banach space are determined. Lastly, the concept of completion of a measure space is generalised for charge spaces, and it is shown that under certain conditions, completion of a charge space adds no new equivalence classes to the quotient space ℒp(X, 𝒜, µ).","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"63 - 89"},"PeriodicalIF":1.0,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44576931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The coarse Ricci curvature of graphs introduced by Ollivier as well as its modification by Lin–Lu– Yau have been studied from various aspects. In this paper, we propose a new transport distance appropriate for hypergraphs and study a generalization of Lin–Lu–Yau type curvature of hypergraphs. As an application, we derive a Bonnet–Myers type estimate for hypergraphs under a lower Ricci curvature bound associated with our transport distance. We remark that our transport distance is new even for graphs and worthy of further study.
{"title":"A New Transport Distance and Its Associated Ricci Curvature of Hypergraphs","authors":"Tomoya Akamatsu","doi":"10.1515/agms-2022-0135","DOIUrl":"https://doi.org/10.1515/agms-2022-0135","url":null,"abstract":"Abstract The coarse Ricci curvature of graphs introduced by Ollivier as well as its modification by Lin–Lu– Yau have been studied from various aspects. In this paper, we propose a new transport distance appropriate for hypergraphs and study a generalization of Lin–Lu–Yau type curvature of hypergraphs. As an application, we derive a Bonnet–Myers type estimate for hypergraphs under a lower Ricci curvature bound associated with our transport distance. We remark that our transport distance is new even for graphs and worthy of further study.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"90 - 108"},"PeriodicalIF":1.0,"publicationDate":"2021-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46125856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Every integral current in a locally compact metric space X can be approximated by a Lipschitz chain with respect to the normal mass, provided that Lipschitz maps into X can be extended slightly.
{"title":"Lipschitz Chain Approximation of Metric Integral Currents","authors":"Tommaso Goldhirsch","doi":"10.1515/agms-2022-0140","DOIUrl":"https://doi.org/10.1515/agms-2022-0140","url":null,"abstract":"Abstract Every integral current in a locally compact metric space X can be approximated by a Lipschitz chain with respect to the normal mass, provided that Lipschitz maps into X can be extended slightly.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"40 - 49"},"PeriodicalIF":1.0,"publicationDate":"2021-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any 0 < β < α, any compact metric space X of Hausdorff dimension α contains a subset which is biLipschitz equivalent to an ultrametric and has Hausdorff dimension at least β. In this note we present a simple proof of the ultrametric skeleton theorem in doubling spaces using Bartal’s Ramsey decompositions [Bartal 2021]. The same general approach is also used to answer a question of Zindulka [Zindulka 2020] about the existence of “nearly ultrametric” subsets of compact spaces having full Hausdorff dimension.
{"title":"A Simple Proof of Dvoretzky-Type Theorem for Hausdorff Dimension in Doubling Spaces","authors":"M. Mendel","doi":"10.1515/agms-2022-0133","DOIUrl":"https://doi.org/10.1515/agms-2022-0133","url":null,"abstract":"Abstract The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any 0 < β < α, any compact metric space X of Hausdorff dimension α contains a subset which is biLipschitz equivalent to an ultrametric and has Hausdorff dimension at least β. In this note we present a simple proof of the ultrametric skeleton theorem in doubling spaces using Bartal’s Ramsey decompositions [Bartal 2021]. The same general approach is also used to answer a question of Zindulka [Zindulka 2020] about the existence of “nearly ultrametric” subsets of compact spaces having full Hausdorff dimension.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"50 - 62"},"PeriodicalIF":1.0,"publicationDate":"2021-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46674203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We propose a model for a growth competition between two subsets of a Riemannian manifold. The sets grow at two different rates, avoiding each other. It is shown that if the competition takes place on a surface which is rotationally symmetric about the starting point of the slower set, then if the surface is conformally equivalent to the Euclidean plane, the slower set remains in a bounded region, while if the surface is nonpositively curved and conformally equivalent to the hyperbolic plane, both sets may keep growing indefinitely.
{"title":"Growth Competitions on Spherically Symmetric Riemannian Manifolds","authors":"Rotem Assouline","doi":"10.1515/agms-2022-0139","DOIUrl":"https://doi.org/10.1515/agms-2022-0139","url":null,"abstract":"Abstract We propose a model for a growth competition between two subsets of a Riemannian manifold. The sets grow at two different rates, avoiding each other. It is shown that if the competition takes place on a surface which is rotationally symmetric about the starting point of the slower set, then if the surface is conformally equivalent to the Euclidean plane, the slower set remains in a bounded region, while if the surface is nonpositively curved and conformally equivalent to the hyperbolic plane, both sets may keep growing indefinitely.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"146 - 154"},"PeriodicalIF":1.0,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study properties of twisted unions of metric spaces introduced in [Johnson, Lindenstrauss, and Schechtman 1986], and in [Naor and Rabani 2017]. In particular, we prove that under certain natural mild assumptions twisted unions of L1-embeddable metric spaces also embed in L1 with distortions bounded above by constants that do not depend on the metric spaces themselves, or on their size, but only on certain general parameters. This answers a question stated in [Naor 2015] and in [Naor and Rabani 2017]. In the second part of the paper we give new simple examples of metric spaces such that their every embedding into Lp, 1 ≤ p < ∞, has distortion at least 3, but which are a union of two subsets, each isometrically embeddable in Lp. This extends the result of [K. Makarychev and Y. Makarychev 2016] from Hilbert spaces to Lp-spaces, 1 ≤ p < ∞.
摘要我们研究了[Johnson，Lindenstrauss，and Schechtman 1986]和[Naor and Rabani 2017]中引入的度量空间的扭并的性质。特别地，我们证明了在某些自然温和的假设下，L1可嵌入度量空间的扭曲并集也嵌入到L1中，其失真由不依赖于度量空间本身或其大小，而仅依赖于某些一般参数的常数所限定。这回答了[Naor 2015]和[Naor和Rabani 2017]中提出的问题。在本文的第二部分中，我们给出了度量空间的新的简单例子，使得它们在Lp，1≤p<∞中的每个嵌入都有至少3的失真，但它们是两个子集的并集，每个子集都可等距嵌入在Lp中。这将[K.Makarychev和Y.Makarychev2016]的结果从Hilbert空间扩展到Lp空间，1≤p<∞。
{"title":"On L1-Embeddability of Unions of L1-Embeddable Metric Spaces and of Twisted Unions of Hypercubes","authors":"M. Ostrovskii, B. Randrianantoanina","doi":"10.1515/agms-2022-0145","DOIUrl":"https://doi.org/10.1515/agms-2022-0145","url":null,"abstract":"Abstract We study properties of twisted unions of metric spaces introduced in [Johnson, Lindenstrauss, and Schechtman 1986], and in [Naor and Rabani 2017]. In particular, we prove that under certain natural mild assumptions twisted unions of L1-embeddable metric spaces also embed in L1 with distortions bounded above by constants that do not depend on the metric spaces themselves, or on their size, but only on certain general parameters. This answers a question stated in [Naor 2015] and in [Naor and Rabani 2017]. In the second part of the paper we give new simple examples of metric spaces such that their every embedding into Lp, 1 ≤ p < ∞, has distortion at least 3, but which are a union of two subsets, each isometrically embeddable in Lp. This extends the result of [K. Makarychev and Y. Makarychev 2016] from Hilbert spaces to Lp-spaces, 1 ≤ p < ∞.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"10 1","pages":"313 - 329"},"PeriodicalIF":1.0,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48049913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.
{"title":"Remarks on Manifolds with Two-Sided Curvature Bounds","authors":"V. Kapovitch, A. Lytchak","doi":"10.1515/agms-2020-0122","DOIUrl":"https://doi.org/10.1515/agms-2020-0122","url":null,"abstract":"Abstract We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"9 1","pages":"53 - 64"},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0122","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46352664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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