The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given $sigma$-compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled. This result has applications in the study of metric Sobolev and BV spaces: it implies that smooth cylindrical functions are dense in energy in these kinds of functional spaces defined over any weighted Banach space.
{"title":"Smooth approximations preserving asymptotic Lipschitz bounds","authors":"Enrico Pasqualetto","doi":"arxiv-2409.01772","DOIUrl":"https://doi.org/arxiv-2409.01772","url":null,"abstract":"The goal of this note is to prove that every real-valued Lipschitz function\u0000on a Banach space can be pointwise approximated on a given $sigma$-compact set\u0000by smooth cylindrical functions whose asymptotic Lipschitz constants are\u0000controlled. This result has applications in the study of metric Sobolev and BV\u0000spaces: it implies that smooth cylindrical functions are dense in energy in\u0000these kinds of functional spaces defined over any weighted Banach space.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"392 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187858","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 complete classification of all zonal, continuous, and translation invariant valuations on convex bodies is established. The valuations obtained are expressed as principal value integrals with respect to the area measures. The convergence of these principal value integrals is obtained from a new weighted version of an inequality for the volume of spherical caps due to Firey. For Minkowski valuations, this implies a refinement of the convolution representation by Schuster and Wannerer in terms of singular integrals. As a further application, a new proof of the classification of $mathrm{SO}(n)$-invariant, continuous, and dually epi-translation invariant valuations on the space of finite convex functions by Colesanti, Ludwig, and Mussnig is obtained.
{"title":"Zonal valuations on convex bodies","authors":"Jonas Knoerr","doi":"arxiv-2409.01897","DOIUrl":"https://doi.org/arxiv-2409.01897","url":null,"abstract":"A complete classification of all zonal, continuous, and translation invariant\u0000valuations on convex bodies is established. The valuations obtained are\u0000expressed as principal value integrals with respect to the area measures. The\u0000convergence of these principal value integrals is obtained from a new weighted\u0000version of an inequality for the volume of spherical caps due to Firey. For\u0000Minkowski valuations, this implies a refinement of the convolution\u0000representation by Schuster and Wannerer in terms of singular integrals. As a\u0000further application, a new proof of the classification of\u0000$mathrm{SO}(n)$-invariant, continuous, and dually epi-translation invariant\u0000valuations on the space of finite convex functions by Colesanti, Ludwig, and\u0000Mussnig is obtained.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187853","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 article, as a first contribution, we provide alternative proofs of recent results by Harrison and Jeffs which determine the precise value of the Gromov-Hausdorff (GH) distance between the circle $mathbb{S}^1$ and the $n$-dimensional sphere $mathbb{S}^n$ (for any $ninmathbb{N}$) when endowed with their respective geodesic metrics. Additionally, we prove that the GH distance between $mathbb{S}^3$ and $mathbb{S}^4$ is equal to $frac{1}{2}arccosleft(frac{-1}{4}right)$, thus settling the case $n=3$ of a conjecture by Lim, M'emoli and Smith.
{"title":"Some novel constructions of optimal Gromov-Hausdorff-optimal correspondences between spheres","authors":"Saúl Rodríguez Martín","doi":"arxiv-2409.02248","DOIUrl":"https://doi.org/arxiv-2409.02248","url":null,"abstract":"In this article, as a first contribution, we provide alternative proofs of\u0000recent results by Harrison and Jeffs which determine the precise value of the\u0000Gromov-Hausdorff (GH) distance between the circle $mathbb{S}^1$ and the\u0000$n$-dimensional sphere $mathbb{S}^n$ (for any $ninmathbb{N}$) when endowed\u0000with their respective geodesic metrics. Additionally, we prove that the GH\u0000distance between $mathbb{S}^3$ and $mathbb{S}^4$ is equal to\u0000$frac{1}{2}arccosleft(frac{-1}{4}right)$, thus settling the case $n=3$ of\u0000a conjecture by Lim, M'emoli and Smith.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187849","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 extend two celebrated inequalities by Busemann -- the random simplex inequality and the intersection inequality -- to both complex and quaternionic vector spaces. Our proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes. Notably, we demonstrate that the standard Steiner symmetrization, contrary to assertions in a paper by Grinberg, does not exhibit this monotonicity property.
{"title":"Complex and Quaternionic Analogues of Busemann's Random Simplex and Intersection Inequalities","authors":"Christos Saroglou, Thomas Wannerer","doi":"arxiv-2409.01057","DOIUrl":"https://doi.org/arxiv-2409.01057","url":null,"abstract":"In this paper, we extend two celebrated inequalities by Busemann -- the\u0000random simplex inequality and the intersection inequality -- to both complex\u0000and quaternionic vector spaces. Our proof leverages a monotonicity property\u0000under symmetrization with respect to complex or quaternionic hyperplanes.\u0000Notably, we demonstrate that the standard Steiner symmetrization, contrary to\u0000assertions in a paper by Grinberg, does not exhibit this monotonicity property.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187854","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 the context of a metric measure space $(X,d,mu)$, we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space $B^theta_{p,p}(X)$ is $k>1$, then $X$ can be decomposed into $k$ number of irreducible components (Theorem 1.1). Note that $theta$ may be bigger than $1$, as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is $1$. We introduce critical exponents $theta_p(X)$ and $theta_p^{ast}(X)$ for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces $X$ formed by glueing copies of $n$-dimensional cubes, the Sierpi'{n}ski gaskets, and of the Sierpi'{n}ski carpet.
{"title":"Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces","authors":"Takashi Kumagai, Nageswari Shanmugalingam, Ryosuke Shimizu","doi":"arxiv-2409.01292","DOIUrl":"https://doi.org/arxiv-2409.01292","url":null,"abstract":"In the context of a metric measure space $(X,d,mu)$, we explore the\u0000potential-theoretic implications of having a finite-dimensional Besov space. We\u0000prove that if the dimension of the Besov space $B^theta_{p,p}(X)$ is $k>1$,\u0000then $X$ can be decomposed into $k$ number of irreducible components (Theorem\u00001.1). Note that $theta$ may be bigger than $1$, as our framework includes\u0000fractals. We also provide sufficient conditions under which the dimension of\u0000the Besov space is $1$. We introduce critical exponents $theta_p(X)$ and\u0000$theta_p^{ast}(X)$ for the Besov spaces. As examples illustrating Theorem\u00001.1, we compute these critical exponents for spaces $X$ formed by glueing\u0000copies of $n$-dimensional cubes, the Sierpi'{n}ski gaskets, and of the\u0000Sierpi'{n}ski carpet.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187679","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}
Building trees to represent or to fit distances is a critical component of phylogenetic analysis, metric embeddings, approximation algorithms, geometric graph neural nets, and the analysis of hierarchical data. Much of the previous algorithmic work, however, has focused on generic metric spaces (i.e., those with no a priori constraints). Leveraging several ideas from the mathematical analysis of hyperbolic geometry and geometric group theory, we study the tree fitting problem as finding the relation between the hyperbolicity (ultrametricity) vector and the error of tree (ultrametric) embedding. That is, we define a vector of hyperbolicity (ultrametric) values over all triples of points and compare the $ell_p$ norms of this vector with the $ell_q$ norm of the distortion of the best tree fit to the distances. This formulation allows us to define the average hyperbolicity (ultrametricity) in terms of a normalized $ell_1$ norm of the hyperbolicity vector. Furthermore, we can interpret the classical tree fitting result of Gromov as a $p = q = infty$ result. We present an algorithm HCCRootedTreeFit such that the $ell_1$ error of the output embedding is analytically bounded in terms of the $ell_1$ norm of the hyperbolicity vector (i.e., $p = q = 1$) and that this result is tight. Furthermore, this algorithm has significantly different theoretical and empirical performance as compared to Gromov's result and related algorithms. Finally, we show using HCCRootedTreeFit and related tree fitting algorithms, that supposedly standard data sets for hierarchical data analysis and geometric graph neural networks have radically different tree fits than those of synthetic, truly tree-like data sets, suggesting that a much more refined analysis of these standard data sets is called for.
{"title":"Fitting trees to $ell_1$-hyperbolic distances","authors":"Joon-Hyeok Yim, Anna C. Gilbert","doi":"arxiv-2409.01010","DOIUrl":"https://doi.org/arxiv-2409.01010","url":null,"abstract":"Building trees to represent or to fit distances is a critical component of\u0000phylogenetic analysis, metric embeddings, approximation algorithms, geometric\u0000graph neural nets, and the analysis of hierarchical data. Much of the previous\u0000algorithmic work, however, has focused on generic metric spaces (i.e., those\u0000with no a priori constraints). Leveraging several ideas from the mathematical\u0000analysis of hyperbolic geometry and geometric group theory, we study the tree\u0000fitting problem as finding the relation between the hyperbolicity\u0000(ultrametricity) vector and the error of tree (ultrametric) embedding. That is,\u0000we define a vector of hyperbolicity (ultrametric) values over all triples of\u0000points and compare the $ell_p$ norms of this vector with the $ell_q$ norm of\u0000the distortion of the best tree fit to the distances. This formulation allows\u0000us to define the average hyperbolicity (ultrametricity) in terms of a\u0000normalized $ell_1$ norm of the hyperbolicity vector. Furthermore, we can\u0000interpret the classical tree fitting result of Gromov as a $p = q = infty$\u0000result. We present an algorithm HCCRootedTreeFit such that the $ell_1$ error\u0000of the output embedding is analytically bounded in terms of the $ell_1$ norm\u0000of the hyperbolicity vector (i.e., $p = q = 1$) and that this result is tight.\u0000Furthermore, this algorithm has significantly different theoretical and\u0000empirical performance as compared to Gromov's result and related algorithms.\u0000Finally, we show using HCCRootedTreeFit and related tree fitting algorithms,\u0000that supposedly standard data sets for hierarchical data analysis and geometric\u0000graph neural networks have radically different tree fits than those of\u0000synthetic, truly tree-like data sets, suggesting that a much more refined\u0000analysis of these standard data sets is called for.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187856","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 $Csubset mathbb{S}^2$ be a spherical convex body of constant width $tau$. It is known that (i) if $tau0$ there exists a spherical convex body $C_varepsilon$ of constant width $tau$ whose boundary consists only of arcs of circles of radius $tau$ such that the Hausdorff distance between $C$ and $C_varepsilon$ is at most $varepsilon$; (ii) if $tau>pi/2$ then for any $varepsilon>0$ there exists a spherical convex body $C_varepsilon$ of constant width $tau$ whose boundary consists only of arcs of circles of radius $tau-frac{pi}{2}$ and great circle arcs such that the Hausdorff distance between $C$ and $C_varepsilon$ is at most $varepsilon$. In this paper, we present an approximation of the remaining case $tau=pi/2$, that is, if $tau=pi/2$ then for any $varepsilon>0$ there exists a spherical polytope $mathcal{P}_varepsilon$ of constant width $pi/2$ such that the Hausdorff distance between $C$ and $mathcal{P}_varepsilon$ is at most $varepsilon$.
{"title":"Approximation of spherical convex bodies of constant width $π/2$","authors":"Huhe Han","doi":"arxiv-2409.00596","DOIUrl":"https://doi.org/arxiv-2409.00596","url":null,"abstract":"Let $Csubset mathbb{S}^2$ be a spherical convex body of constant width\u0000$tau$. It is known that (i) if $tau<pi/2$ then for any $varepsilon>0$ there\u0000exists a spherical convex body $C_varepsilon$ of constant width $tau$ whose\u0000boundary consists only of arcs of circles of radius $tau$ such that the\u0000Hausdorff distance between $C$ and $C_varepsilon$ is at most $varepsilon$;\u0000(ii) if $tau>pi/2$ then for any $varepsilon>0$ there exists a spherical\u0000convex body $C_varepsilon$ of constant width $tau$ whose boundary consists\u0000only of arcs of circles of radius $tau-frac{pi}{2}$ and great circle arcs\u0000such that the Hausdorff distance between $C$ and $C_varepsilon$ is at most\u0000$varepsilon$. In this paper, we present an approximation of the remaining case\u0000$tau=pi/2$, that is, if $tau=pi/2$ then for any $varepsilon>0$ there\u0000exists a spherical polytope $mathcal{P}_varepsilon$ of constant width $pi/2$\u0000such that the Hausdorff distance between $C$ and $mathcal{P}_varepsilon$ is\u0000at most $varepsilon$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"2022 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187855","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}
Recently, Greenfeld and Tao disproof the conjecture that translational tilings of a single tile can always be periodic [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, the translational tiling for subsets of $mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper shows that translational tiling of the $3$-dimensional space with a set of $5$ polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from $3$-dimensional space to $4$-dimensional space, we manage to show that translational tiling of the $4$-dimensional space with a set of $4$ tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the $n$-dimensional space with a monotile, for some fixed $n$.
{"title":"Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes","authors":"Chao Yang, Zhujun Zhang","doi":"arxiv-2409.00846","DOIUrl":"https://doi.org/arxiv-2409.00846","url":null,"abstract":"Recently, Greenfeld and Tao disproof the conjecture that translational\u0000tilings of a single tile can always be periodic [Ann. Math. 200(2024),\u0000301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show\u0000that if the dimension $n$ is part of the input, the translational tiling for\u0000subsets of $mathbb{Z}^n$ with one tile is undecidable. These two results are\u0000very strong pieces of evidence for the conjecture that translational tiling of\u0000$mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper\u0000shows that translational tiling of the $3$-dimensional space with a set of $5$\u0000polycubes is undecidable. By introducing a technique that lifts a set of\u0000polycubes and its tiling from $3$-dimensional space to $4$-dimensional space,\u0000we manage to show that translational tiling of the $4$-dimensional space with a\u0000set of $4$ tiles is undecidable. This is a step towards the attempt to settle\u0000the conjecture of the undecidability of translational tiling of the\u0000$n$-dimensional space with a monotile, for some fixed $n$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187857","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 tiling is said to have infinite local complexity (ILC) if it contains infinitely many two-tile patches up to rigid motions. In this work, we provide examples of substitution rules that generate tilings with ILC. The proof relies on Danzer's algorithm, which assumes that the substitution factor is non-Pisot. In addition to ILC, the tiling space of each substitution rule contains a tiling that exhibits global n-fold rotational symmetry, n=13,17,21.
如果一个瓦片包含无限多的双瓦片补丁，直到刚性运动为止，那么这个瓦片就被称为具有无限局部复杂性（ILC）。在这项工作中，我们提供了生成具有 ILC 的平铺的置换规则实例。除了 ILC 之外，每个替换规则的平铺空间还包含具有全局 n 倍旋转对称性（n=13,17,21）的平铺。
{"title":"Tilings with Infinite Local Complexity and n-Fold Rotational Symmetry, n=13,17,21","authors":"April Lynne D. Say-awen","doi":"arxiv-2408.17082","DOIUrl":"https://doi.org/arxiv-2408.17082","url":null,"abstract":"A tiling is said to have infinite local complexity (ILC) if it contains\u0000infinitely many two-tile patches up to rigid motions. In this work, we provide\u0000examples of substitution rules that generate tilings with ILC. The proof relies\u0000on Danzer's algorithm, which assumes that the substitution factor is non-Pisot.\u0000In addition to ILC, the tiling space of each substitution rule contains a\u0000tiling that exhibits global n-fold rotational symmetry, n=13,17,21.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187860","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 major problem in analysis on metric spaces to understand when a metric space is quasisymmetric to a space with strong analytic structure, a so-called Loewner space. A conjecture of Kleiner, recently disproven by Anttila and the second author, proposes a combinatorial sufficient condition. The counterexamples constructed are all topologically one dimensional, and the sufficiency of Kleiner's condition remains open for most other examples. A separate question of Kleiner and Schioppa, apparently unrelated to the problem above, asks about the existence of "analytically $1$-dimensional planes": metric measure spaces quasisymmetric to the Euclidean plane but supporting a $1$-dimensional analytic structure in the sense of Cheeger. In this paper, we construct an example for which the conclusion of Kleiner's conjecture is not known to hold. We show that either this conclusion fails in our example or there exists an "analytically $1$-dimensional plane". Thus, our construction either yields a new counterexample to Kleiner's conjecture, different in kind from those of Anttila and the second author, or a resolution to the problem of Kleiner--Schioppa.
{"title":"Analytically one-dimensional planes and the Combinatorial Loewner Property","authors":"Guy C. David, Sylvester Eriksson-Bique","doi":"arxiv-2408.17279","DOIUrl":"https://doi.org/arxiv-2408.17279","url":null,"abstract":"It is a major problem in analysis on metric spaces to understand when a\u0000metric space is quasisymmetric to a space with strong analytic structure, a\u0000so-called Loewner space. A conjecture of Kleiner, recently disproven by Anttila\u0000and the second author, proposes a combinatorial sufficient condition. The\u0000counterexamples constructed are all topologically one dimensional, and the\u0000sufficiency of Kleiner's condition remains open for most other examples. A separate question of Kleiner and Schioppa, apparently unrelated to the\u0000problem above, asks about the existence of \"analytically $1$-dimensional\u0000planes\": metric measure spaces quasisymmetric to the Euclidean plane but\u0000supporting a $1$-dimensional analytic structure in the sense of Cheeger. In this paper, we construct an example for which the conclusion of Kleiner's\u0000conjecture is not known to hold. We show that either this conclusion fails in\u0000our example or there exists an \"analytically $1$-dimensional plane\". Thus, our\u0000construction either yields a new counterexample to Kleiner's conjecture,\u0000different in kind from those of Anttila and the second author, or a resolution\u0000to the problem of Kleiner--Schioppa.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187859","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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