The Favard length of a Borel set $Esubsetmathbb{R}^2$ is the average length of its orthogonal projections. We prove that if $E$ is Ahlfors 1-regular and it has large Favard length, then it contains a big piece of a Lipschitz graph. This gives a quantitative version of the Besicovitch projection theorem. As a corollary, we answer questions of David and Semmes and of Peres and Solomyak. We also make progress on Vitushkin's conjecture.
{"title":"Favard length and quantitative rectifiability","authors":"Damian Dąbrowski","doi":"arxiv-2408.03919","DOIUrl":"https://doi.org/arxiv-2408.03919","url":null,"abstract":"The Favard length of a Borel set $Esubsetmathbb{R}^2$ is the average length\u0000of its orthogonal projections. We prove that if $E$ is Ahlfors 1-regular and it\u0000has large Favard length, then it contains a big piece of a Lipschitz graph.\u0000This gives a quantitative version of the Besicovitch projection theorem. As a\u0000corollary, we answer questions of David and Semmes and of Peres and Solomyak.\u0000We also make progress on Vitushkin's conjecture.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934875","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}
Kleiner-Leeb and Quint showed that convex subsets in higher-rank symmetric spaces are very rigid compared to rank 1 symmetric spaces. Motivated by this, we consider convex subsets in products of proper CAT(0) spaces $X_1times X_2$ and show that for any two convex co-compact actions $rho_i(Gamma)$ on $X_i$, where $i=1, 2$, if the diagonal action of $Gamma$ on $X_1times X_2$ via $rho=(rho_1, rho_2)$ is also convex co-compact, then under a suitable condition, $rho_1(Gamma)$ and $rho_2(Gamma)$ have the same marked length spectrum.
{"title":"Rigidity of convex co-compact diagonal actions","authors":"Subhadip Dey, Beibei Liu","doi":"arxiv-2408.03462","DOIUrl":"https://doi.org/arxiv-2408.03462","url":null,"abstract":"Kleiner-Leeb and Quint showed that convex subsets in higher-rank symmetric\u0000spaces are very rigid compared to rank 1 symmetric spaces. Motivated by this,\u0000we consider convex subsets in products of proper CAT(0) spaces $X_1times X_2$\u0000and show that for any two convex co-compact actions $rho_i(Gamma)$ on $X_i$,\u0000where $i=1, 2$, if the diagonal action of $Gamma$ on $X_1times X_2$ via\u0000$rho=(rho_1, rho_2)$ is also convex co-compact, then under a suitable\u0000condition, $rho_1(Gamma)$ and $rho_2(Gamma)$ have the same marked length\u0000spectrum.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"112 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934876","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}
An integral curve is a closed piecewise linear curve comprised of unit intervals. A dome is a polyhedral surface whose faces are equilateral triangles and whose boundary is an integral curve. Glazyrin and Pak showed that not every integral curve can be domed by analyzing the case of unit rhombi, and conjectured that every integral curve is cobordant to a unit rhombus. We show that this is false for oriented domes, but that every integral curve is cobordant to the union of finitely many unit rhombi.
积分曲线是由单位区间组成的封闭的片断线性曲线。穹顶是面为等边三角形、边界为积分曲线的多面体。Glazyrin 和 Pak 通过分析单位菱形的情况，证明并非每条积分曲线都能形成穹顶，并推测每条积分曲线都与单位菱形共线。我们证明了这一推测对于有向圆顶来说是错误的，但每条积分曲线都与有限多个单位菱形的联合体共线。
{"title":"Cobordism of domes over curves","authors":"Robert Miranda","doi":"arxiv-2408.02517","DOIUrl":"https://doi.org/arxiv-2408.02517","url":null,"abstract":"An integral curve is a closed piecewise linear curve comprised of unit\u0000intervals. A dome is a polyhedral surface whose faces are equilateral triangles\u0000and whose boundary is an integral curve. Glazyrin and Pak showed that not every\u0000integral curve can be domed by analyzing the case of unit rhombi, and\u0000conjectured that every integral curve is cobordant to a unit rhombus. We show\u0000that this is false for oriented domes, but that every integral curve is\u0000cobordant to the union of finitely many unit rhombi.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934877","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 focuses on the undecidability of translational tiling of $n$-dimensional space $mathbb{Z}^n$ with a set of $k$ tiles. It is known that tiling $mathbb{Z}^2$ with translated copies with a set of $8$ tiles is undecidable. Greenfeld and Tao gave strong evidence in a series of works that for sufficiently large dimension $n$, the translational tiling problem for $mathbb{Z}^n$ might be undecidable for just one tile. This paper shows the undecidability of translational tiling of $mathbb{Z}^3$ with a set of $6$ tiles.
{"title":"Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes","authors":"Chao Yang, Zhujun Zhang","doi":"arxiv-2408.02196","DOIUrl":"https://doi.org/arxiv-2408.02196","url":null,"abstract":"This paper focuses on the undecidability of translational tiling of\u0000$n$-dimensional space $mathbb{Z}^n$ with a set of $k$ tiles. It is known that\u0000tiling $mathbb{Z}^2$ with translated copies with a set of $8$ tiles is\u0000undecidable. Greenfeld and Tao gave strong evidence in a series of works that\u0000for sufficiently large dimension $n$, the translational tiling problem for\u0000$mathbb{Z}^n$ might be undecidable for just one tile. This paper shows the\u0000undecidability of translational tiling of $mathbb{Z}^3$ with a set of $6$\u0000tiles.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934879","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 consider the setting of a locally compact, non-complete metric measure space $(Z,d,nu)$ equipped with a doubling measure $nu$, under the condition that the boundary $partial Z:=overline{Z}setminus Z$ (obtained by considering the completion of $Z$) supports a Radon measure $pi$ which is in a $sigma$-codimensional relationship to $nu$ for some $sigma>0$. We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on $Z$. We also establish interior regularity of solutions when the inhomogeneity data is in an $L^q$-class for sufficiently large $q>1$, and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous.
{"title":"Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces","authors":"Josh Kline, Feng Li, Nageswari Shanmugalingam","doi":"arxiv-2408.02624","DOIUrl":"https://doi.org/arxiv-2408.02624","url":null,"abstract":"In this paper we consider the setting of a locally compact, non-complete\u0000metric measure space $(Z,d,nu)$ equipped with a doubling measure $nu$, under\u0000the condition that the boundary $partial Z:=overline{Z}setminus Z$ (obtained\u0000by considering the completion of $Z$) supports a Radon measure $pi$ which is\u0000in a $sigma$-codimensional relationship to $nu$ for some $sigma>0$. We\u0000explore existence, uniqueness, comparison property, and stability properties of\u0000solutions to inhomogeneous Dirichlet problems associated with certain nonlinear\u0000nonlocal operators on $Z$. We also establish interior regularity of solutions\u0000when the inhomogeneity data is in an $L^q$-class for sufficiently large $q>1$,\u0000and verify a Kellogg-type property when the inhomogeneity data vanishes and the\u0000Dirichlet data is continuous.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"195 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934878","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 show that any finite family of pairwise intersecting balls in $mathbb{E}^n$ can be pierced by $(sqrt{3/2}+o(1))^n$ points improving the previously known estimate of $(2+o(1))^n$. As a corollary, this implies that any $2$-illuminable spiky ball in $mathbb{E}^n$ can be illuminated by $(sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky balls, i.e., cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of $1.19851^n$, which can be compared with the previous $(sqrt{2}+o(1))^n$ established only for the centrally symmetric cap bodies. We also prove the lower bounds of $(tfrac{2}{sqrt{3}}-o(1))^n$ for the three problems above.
{"title":"On a Gallai-type problem and illumination of spiky balls and cap bodies","authors":"Andrii Arman, Andriy Bondarenko, Andriy Prymak, Danylo Radchenko","doi":"arxiv-2408.01341","DOIUrl":"https://doi.org/arxiv-2408.01341","url":null,"abstract":"We show that any finite family of pairwise intersecting balls in\u0000$mathbb{E}^n$ can be pierced by $(sqrt{3/2}+o(1))^n$ points improving the\u0000previously known estimate of $(2+o(1))^n$. As a corollary, this implies that\u0000any $2$-illuminable spiky ball in $mathbb{E}^n$ can be illuminated by\u0000$(sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky\u0000balls, i.e., cap bodies, we show an upper bound in terms of the sizes of\u0000certain related spherical codes and coverings. For large dimensions, this\u0000results in an upper bound of $1.19851^n$, which can be compared with the\u0000previous $(sqrt{2}+o(1))^n$ established only for the centrally symmetric cap\u0000bodies. We also prove the lower bounds of $(tfrac{2}{sqrt{3}}-o(1))^n$ for\u0000the three problems above.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934880","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}
Valuations on the space of finite-valued convex functions on $mathbb{C}^n$ that are continuous, dually epi-translation invariant, as well as $mathrm{U}(n)$-invariant are completely classified. It is shown that the space of these valuations decomposes into a direct sum of subspaces defined in terms of vanishing properties with respect to restrictions to a finite family of special subspaces of $mathbb{C}^n$, mirroring the behavior of the hermitian intrinsic volumes introduced by Bernig and Fu. Unique representations of these valuations in terms of principal value integrals involving two families of Monge-Amp`ere-type operators are established
{"title":"A geometric decomposition for unitarily invariant valuations on convex functions","authors":"Jonas Knoerr","doi":"arxiv-2408.01352","DOIUrl":"https://doi.org/arxiv-2408.01352","url":null,"abstract":"Valuations on the space of finite-valued convex functions on $mathbb{C}^n$\u0000that are continuous, dually epi-translation invariant, as well as\u0000$mathrm{U}(n)$-invariant are completely classified. It is shown that the space\u0000of these valuations decomposes into a direct sum of subspaces defined in terms\u0000of vanishing properties with respect to restrictions to a finite family of\u0000special subspaces of $mathbb{C}^n$, mirroring the behavior of the hermitian\u0000intrinsic volumes introduced by Bernig and Fu. Unique representations of these\u0000valuations in terms of principal value integrals involving two families of\u0000Monge-Amp`ere-type operators are established","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934883","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 generalise an interesting geometry problem from the 1995 edition of the International Mathematical Olympiad (IMO) using analytic geometry tools.
{"title":"Generalisation of an IMO Geometry Problem","authors":"Dina Kamber Hamzić, László Németh, Zenan Šabanac","doi":"arxiv-2408.01071","DOIUrl":"https://doi.org/arxiv-2408.01071","url":null,"abstract":"In this paper, we generalise an interesting geometry problem from the 1995\u0000edition of the International Mathematical Olympiad (IMO) using analytic\u0000geometry tools.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934881","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}
Davi Lopes Medeiros, José Edson Sampaio, Emanoel Souza
The Moderately Discontinuous Homology (MD-Homology, for short) was created recently in 2022 by Fern'andez de Bobadilla at al. and it captures deep Lipschitz phenomena. However, to become a definitive powerful tool, it must be widely comprehended. In this paper, we investigate the MD-Homology of definable surface germs for the inner and outer metrics. We completely determine the MD-Homology of surfaces for the inner metric and we present a great variety of interesting MD-Homology of surfaces for the outer metric, for instance, we determine the MD-Homology of some bubbles, snake surfaces, and horns. Furthermore, we explicit the diversity of MD-Homology of surfaces for the outer metric in general, showing how hard it is to completely solve the outer classification problem. On the other hand, we show that, under specific conditions, the weakly outer Lipschitz equivalence determines completely the MD-Homology of surfaces for the outer metric, showing that these two subjects are quite related.
{"title":"Moderately discontinuous homology of real surfaces","authors":"Davi Lopes Medeiros, José Edson Sampaio, Emanoel Souza","doi":"arxiv-2408.00851","DOIUrl":"https://doi.org/arxiv-2408.00851","url":null,"abstract":"The Moderately Discontinuous Homology (MD-Homology, for short) was created\u0000recently in 2022 by Fern'andez de Bobadilla at al. and it captures deep\u0000Lipschitz phenomena. However, to become a definitive powerful tool, it must be\u0000widely comprehended. In this paper, we investigate the MD-Homology of definable surface germs for\u0000the inner and outer metrics. We completely determine the MD-Homology of\u0000surfaces for the inner metric and we present a great variety of interesting\u0000MD-Homology of surfaces for the outer metric, for instance, we determine the\u0000MD-Homology of some bubbles, snake surfaces, and horns. Furthermore, we\u0000explicit the diversity of MD-Homology of surfaces for the outer metric in\u0000general, showing how hard it is to completely solve the outer classification\u0000problem. On the other hand, we show that, under specific conditions, the weakly\u0000outer Lipschitz equivalence determines completely the MD-Homology of surfaces\u0000for the outer metric, showing that these two subjects are quite related.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934882","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 focuses on studying the configuration spaces of graphs realised in $mathbb C^2$, such that the configuration space is, after normalisation, one dimensional. If this is the case, then the configuration space is, generically, a smooth complex curve, and can be seen as a Riemann surface. The property of interest in this paper is the genus of this curve. Using tropical geometry, we give an algorithm to compute this genus. We provide an implementation in Python and give various examples.
{"title":"On the Genus of One Degree of Freedom Planar Linkages via Tropical Geometry","authors":"Josef Schicho, Ayush Kumar Tewari, Audie Warren","doi":"arxiv-2408.00449","DOIUrl":"https://doi.org/arxiv-2408.00449","url":null,"abstract":"This paper focuses on studying the configuration spaces of graphs realised in\u0000$mathbb C^2$, such that the configuration space is, after normalisation, one\u0000dimensional. If this is the case, then the configuration space is, generically,\u0000a smooth complex curve, and can be seen as a Riemann surface. The property of\u0000interest in this paper is the genus of this curve. Using tropical geometry, we\u0000give an algorithm to compute this genus. We provide an implementation in Python\u0000and give various examples.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882945","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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