If we consider a sequence of warped product length spaces, what conditions on the sequence of warping functions implies compactness of the sequence of distance functions? In particular, we want to know when a subsequence converges to a well defined metric space on the same manifold with the same topology. What conditions on the sequence of warping functions implies Lipschitz bounds for the sequence of distance functions and/or the limiting distance function? In this paper we give answers to both of these questions as well as many examples which elucidate the theorems and show that our hypotheses are necessary.
{"title":"Compactness of Sequences of Warped Product Length Spaces","authors":"Brian Allen, Bryan Sanchez, Yahaira Torres","doi":"arxiv-2409.07193","DOIUrl":"https://doi.org/arxiv-2409.07193","url":null,"abstract":"If we consider a sequence of warped product length spaces, what conditions on\u0000the sequence of warping functions implies compactness of the sequence of\u0000distance functions? In particular, we want to know when a subsequence converges\u0000to a well defined metric space on the same manifold with the same topology.\u0000What conditions on the sequence of warping functions implies Lipschitz bounds\u0000for the sequence of distance functions and/or the limiting distance function?\u0000In this paper we give answers to both of these questions as well as many\u0000examples which elucidate the theorems and show that our hypotheses are\u0000necessary.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187833","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 $p_1,ldots,p_n$ be a set of points in the unit square and let $T_1,ldots,T_n$ be a set of $delta$-tubes such that $T_j$ passes through $p_j$. We prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, we show that in any configuration of points $p_1,ldots, p_n in [0,1]^2$ along with a line $ell_j$ through each point $p_j$, there exist $jneq k$ for which $d(p_j, ell_k) lesssim n^{-2/3+o(1)}$. It follows from the latter result that any set of $n$ points in the unit square contains three points forming a triangle of area at most $n^{-7/6+o(1)}$. This new upper bound for Heilbronn's triangle problem attains the high-low limit established in our previous work arXiv:2305.18253.
{"title":"Lower bounds for incidences","authors":"Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov","doi":"arxiv-2409.07658","DOIUrl":"https://doi.org/arxiv-2409.07658","url":null,"abstract":"Let $p_1,ldots,p_n$ be a set of points in the unit square and let\u0000$T_1,ldots,T_n$ be a set of $delta$-tubes such that $T_j$ passes through\u0000$p_j$. We prove a lower bound for the number of incidences between the points\u0000and tubes under a natural regularity condition (similar to Frostman\u0000regularity). As a consequence, we show that in any configuration of points\u0000$p_1,ldots, p_n in [0,1]^2$ along with a line $ell_j$ through each point\u0000$p_j$, there exist $jneq k$ for which $d(p_j, ell_k) lesssim n^{-2/3+o(1)}$. It follows from the latter result that any set of $n$ points in the unit\u0000square contains three points forming a triangle of area at most\u0000$n^{-7/6+o(1)}$. This new upper bound for Heilbronn's triangle problem attains\u0000the high-low limit established in our previous work arXiv:2305.18253.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187831","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 thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the $d$-dimensional sphere $S^d$. In paper I we present some facts describing the shape of reduced bodies of thickness under $frac{pi}{2}$ on $S^2$. We also consider reduced bodies of thickness at least $frac{pi}{2}$, which appear to be of constant width. Paper II focuses on bodies of constant width on $S^d$. We present the properties of these bodies and in particular we discuss conections between notions of constant width and of constant diameter. In paper III we estimate the diameter of a reduced convex body. The main theme of paper IV is estimating the radius of the smallest disk that covers a reduced convex body on $S^2$. The result of paper V is showing that every spherical reduced polygon $V$ is contained in a disk of radius equal to the thickness of this body centered at a boundary point of $V$.
本论文由五篇论文组成，涉及还原球形凸体，特别是 $d$ 维球面$S^d$上的恒宽球形体。在论文 I 中，我们提出了一些描述在 $S^2$ 上 $frac{pi}{2}$ 下厚度减小体形状的事实。我们还考虑了厚度至少为 $frac{pi}{2}$ 的还原体，它们看起来宽度不变。论文二的重点是$S^d$上的恒宽体。我们介绍了这些体的性质，特别是讨论了恒定宽度与恒定直径概念之间的联系。在论文 III 中，我们估计了还原凸体的直径。论文 IV 的主题是估计覆盖 $S^2$ 上还原凸体的最小圆盘的半径。论文 V 的结果表明，每一个球形还原多边形 $V$ 都包含在以 $V$ 边界点为中心的半径等于该体厚度的圆盘中。
{"title":"On reduced spherical bodies","authors":"Michał Musielak","doi":"arxiv-2409.07036","DOIUrl":"https://doi.org/arxiv-2409.07036","url":null,"abstract":"This thesis consists of five papers about reduced spherical convex bodies and\u0000in particular spherical bodies of constant width on the $d$-dimensional sphere\u0000$S^d$. In paper I we present some facts describing the shape of reduced bodies\u0000of thickness under $frac{pi}{2}$ on $S^2$. We also consider reduced bodies of\u0000thickness at least $frac{pi}{2}$, which appear to be of constant width. Paper\u0000II focuses on bodies of constant width on $S^d$. We present the properties of\u0000these bodies and in particular we discuss conections between notions of\u0000constant width and of constant diameter. In paper III we estimate the diameter\u0000of a reduced convex body. The main theme of paper IV is estimating the radius\u0000of the smallest disk that covers a reduced convex body on $S^2$. The result of\u0000paper V is showing that every spherical reduced polygon $V$ is contained in a\u0000disk of radius equal to the thickness of this body centered at a boundary point\u0000of $V$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187832","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}
For the action of the orthogonal group or euclidean group on k-tuples of vectors we construct a bi-Lipschitz embedding from the orbit space into euclidean space.This embedding has distortion sqrt(2).
对于正交群或欧几里得群对 k 个向量元组的作用，我们构建了一个从轨道空间到欧几里得空间的双利普希茨嵌入。
{"title":"Bi-Lipschitz Quotient embedding for Euclidean Group actions on Data","authors":"Harm Derksen","doi":"arxiv-2409.06829","DOIUrl":"https://doi.org/arxiv-2409.06829","url":null,"abstract":"For the action of the orthogonal group or euclidean group on k-tuples of\u0000vectors we construct a bi-Lipschitz embedding from the orbit space into\u0000euclidean space.This embedding has distortion sqrt(2).","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187834","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 obtain the Lipschitz analogues of the results Perelman used from Siebenmann's deformation of homeomorphism theory in his proof of the stability theorem. Consequently, we obtain the Lipschitz analogue of Perelman's gluing theorem. Moreover, we obtain the analogous deformation theory but with tracking of the Lipschitz constants.
{"title":"Deformations of Lipschitz Homeomorphisms","authors":"Mohammad Alattar","doi":"arxiv-2409.06170","DOIUrl":"https://doi.org/arxiv-2409.06170","url":null,"abstract":"We obtain the Lipschitz analogues of the results Perelman used from\u0000Siebenmann's deformation of homeomorphism theory in his proof of the stability\u0000theorem. Consequently, we obtain the Lipschitz analogue of Perelman's gluing\u0000theorem. Moreover, we obtain the analogous deformation theory but with tracking\u0000of the Lipschitz constants.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187839","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 introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is a special case of the master integrable equation Q4. The variational description is given in terms of elliptic generalizations of the dilogarithm function. They have the same convexity principles as their circle-pattern counterparts. This allows us to prove existence and uniqueness results for the Dirichlet and Neumann boundary value problems. Some examples are computed numerically. In the limit of small smoothly varying rings, one obtains harmonic maps to the sphere and to the hyperbolic plane. A close relation to discrete surfaces with constant mean curvature is explained.
{"title":"Spherical and hyperbolic orthogonal ring patterns: integrability and variational principles","authors":"Alexander I. Bobenko","doi":"arxiv-2409.06573","DOIUrl":"https://doi.org/arxiv-2409.06573","url":null,"abstract":"We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic\u0000plane, consisting of pairs of concentric circles, which generalize circle\u0000patterns. We show that their radii are described by a discrete integrable\u0000system. This is a special case of the master integrable equation Q4. The\u0000variational description is given in terms of elliptic generalizations of the\u0000dilogarithm function. They have the same convexity principles as their\u0000circle-pattern counterparts. This allows us to prove existence and uniqueness\u0000results for the Dirichlet and Neumann boundary value problems. Some examples\u0000are computed numerically. In the limit of small smoothly varying rings, one\u0000obtains harmonic maps to the sphere and to the hyperbolic plane. A close\u0000relation to discrete surfaces with constant mean curvature is explained.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187836","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 a classical fact in Euclidean geometry that the regular polygon maximizes area amongst polygons of the same perimeter and number of sides, and the analogue of this in non-Euclidean geometries has long been a folklore result. In this note, we present a complete proof of this polygonal isoperimetric inequality in hyperbolic and spherical geometries.
{"title":"Isoperimetric inequality for non-Euclidean polygons","authors":"Basudeb Datta, Subhojoy Gupta","doi":"arxiv-2409.06529","DOIUrl":"https://doi.org/arxiv-2409.06529","url":null,"abstract":"It is a classical fact in Euclidean geometry that the regular polygon\u0000maximizes area amongst polygons of the same perimeter and number of sides, and\u0000the analogue of this in non-Euclidean geometries has long been a folklore\u0000result. In this note, we present a complete proof of this polygonal\u0000isoperimetric inequality in hyperbolic and spherical geometries.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187837","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}
Intermediate dimensions are a class of new fractal dimensions which provide a spectrum of dimensions interpolating between the Hausdorff and box-counting dimensions. In this paper, we study the intermediate dimensions of Moran sets. Moran sets may be regarded as a generalization of self-similar sets generated by using different class of similar mappings at each level with unfixed translations, and this causes the lack of ergodic properties on Moran set. Therefore, the intermediate dimensions do not necessarily exist, and we calculate the upper and lower intermediate dimensions of Moran sets. In particular, we obtain a simplified intermediate dimension formula for homogeneous Moran sets. Moreover, we study the visualization of the upper intermediate dimensions for some homogeneous Moran sets, and we show that their upper intermediate dimensions are given by Mobius transformations.
{"title":"Intermediate dimensions of Moran sets and their visualization","authors":"Yali Du, Junjie Miao, Tianrui Wang, Haojie Xu","doi":"arxiv-2409.06186","DOIUrl":"https://doi.org/arxiv-2409.06186","url":null,"abstract":"Intermediate dimensions are a class of new fractal dimensions which provide a\u0000spectrum of dimensions interpolating between the Hausdorff and box-counting\u0000dimensions. In this paper, we study the intermediate dimensions of Moran sets. Moran sets\u0000may be regarded as a generalization of self-similar sets generated by using\u0000different class of similar mappings at each level with unfixed translations,\u0000and this causes the lack of ergodic properties on Moran set. Therefore, the\u0000intermediate dimensions do not necessarily exist, and we calculate the upper\u0000and lower intermediate dimensions of Moran sets. In particular, we obtain a\u0000simplified intermediate dimension formula for homogeneous Moran sets. Moreover,\u0000we study the visualization of the upper intermediate dimensions for some\u0000homogeneous Moran sets, and we show that their upper intermediate dimensions\u0000are given by Mobius transformations.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187838","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}
Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santal'o inequality in its functional form. Inspired by their method, based on the Fokker-Planck semi-group, we extend the inequality to non-even functions. We consider a well-chosen centering procedure by studying the infimum over translations in a double Laplace transform. This requires a new look on the existing methods and leads to several observations of independent interest on the geometry of the Laplace transform. Application to reverse hypercontractivity is also given.
{"title":"On a Santaló point for Nakamura-Tsuji's Laplace transform inequality","authors":"Dario Cordero-Erausquin, Matthieu Fradelizi, Dylan Langharst","doi":"arxiv-2409.05541","DOIUrl":"https://doi.org/arxiv-2409.05541","url":null,"abstract":"Nakamura and Tsuji recently obtained an integral inequality involving a\u0000Laplace transform of even functions that implies, at the limit, the\u0000Blaschke-Santal'o inequality in its functional form. Inspired by their method,\u0000based on the Fokker-Planck semi-group, we extend the inequality to non-even\u0000functions. We consider a well-chosen centering procedure by studying the\u0000infimum over translations in a double Laplace transform. This requires a new\u0000look on the existing methods and leads to several observations of independent\u0000interest on the geometry of the Laplace transform. Application to reverse\u0000hypercontractivity is also given.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187841","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 the coarse kernel of a group action, namely the normal subgroup of elements that translate every point by a uniformly bounded amount. We give a complete algebraic characterization of this object. We specialize to $mathrm{CAT}(0)$ spaces and show that the coarse kernel must be virtually abelian, characterizing when it is finite or cyclic in terms of the curtain model. As an application, we characterize the relation between the coarse kernels of the action on a $mathrm{CAT}(0)$ space and the induced action on its curtain model. Along the way, we study weakly acylindrical actions on quasi-lines.
{"title":"Coarse Kernels of Group Actions","authors":"Tejas Mittal","doi":"arxiv-2409.05288","DOIUrl":"https://doi.org/arxiv-2409.05288","url":null,"abstract":"In this paper, we study the coarse kernel of a group action, namely the\u0000normal subgroup of elements that translate every point by a uniformly bounded\u0000amount. We give a complete algebraic characterization of this object. We\u0000specialize to $mathrm{CAT}(0)$ spaces and show that the coarse kernel must be\u0000virtually abelian, characterizing when it is finite or cyclic in terms of the\u0000curtain model. As an application, we characterize the relation between the\u0000coarse kernels of the action on a $mathrm{CAT}(0)$ space and the induced\u0000action on its curtain model. Along the way, we study weakly acylindrical\u0000actions on quasi-lines.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187840","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：