An iterated circumcenter sequence (ICS) in dimension $d$ is a sequence of points in $mathbb{R}^d$ where each point is the circumcenter of the preceding $d+1$ points. The purpose of this paper is to completely determine the parameter space of ICSs and its subspace consisting of periodic ICSs. In particular, we prove Goddyn's conjecture on periodic ICSs, which was independently proven recently by Ardanuy. We also prove the existence of a periodic ICS in any dimension.
{"title":"On iterated circumcenter sequences","authors":"Shuho Kanda, Junnosuke Koizumi","doi":"arxiv-2407.19767","DOIUrl":"https://doi.org/arxiv-2407.19767","url":null,"abstract":"An iterated circumcenter sequence (ICS) in dimension $d$ is a sequence of\u0000points in $mathbb{R}^d$ where each point is the circumcenter of the preceding\u0000$d+1$ points. The purpose of this paper is to completely determine the\u0000parameter space of ICSs and its subspace consisting of periodic ICSs. In\u0000particular, we prove Goddyn's conjecture on periodic ICSs, which was\u0000independently proven recently by Ardanuy. We also prove the existence of a\u0000periodic ICS in any dimension.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870127","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 metric polygon is a metric space comprised of a finite number of closed intervals joined cyclically. The second-named author and Ntalampekos recently found a method to bi-Lipschitz embed an arbitrary metric triangle in the Euclidean plane with uniformly bounded distortion, which we call here the tripodal embedding. In this paper, we prove the sharp distortion bound $4sqrt{7/3}$ for the tripodal embedding. We also give a detailed analysis of four representative examples of metric triangles: the intrinsic circle, the three-petal rose, tripods and the twisted heart. In particular, our examples show the sharpness of the tripodal embedding distortion bound and give a lower bound for the optimal distortion bound in general. Finally, we show the triangle embedding theorem does not generalize to metric quadrilaterals by giving a family of examples of metric quadrilaterals that are not bi-Lipschitz embeddable in the plane with uniform distortion.
{"title":"Bi-Lipschitz embedding metric triangles in the plane","authors":"Xinyuan Luo, Matthew Romney, Alexandria L. Tao","doi":"arxiv-2407.20019","DOIUrl":"https://doi.org/arxiv-2407.20019","url":null,"abstract":"A metric polygon is a metric space comprised of a finite number of closed\u0000intervals joined cyclically. The second-named author and Ntalampekos recently\u0000found a method to bi-Lipschitz embed an arbitrary metric triangle in the\u0000Euclidean plane with uniformly bounded distortion, which we call here the\u0000tripodal embedding. In this paper, we prove the sharp distortion bound\u0000$4sqrt{7/3}$ for the tripodal embedding. We also give a detailed analysis of\u0000four representative examples of metric triangles: the intrinsic circle, the\u0000three-petal rose, tripods and the twisted heart. In particular, our examples\u0000show the sharpness of the tripodal embedding distortion bound and give a lower\u0000bound for the optimal distortion bound in general. Finally, we show the\u0000triangle embedding theorem does not generalize to metric quadrilaterals by\u0000giving a family of examples of metric quadrilaterals that are not bi-Lipschitz\u0000embeddable in the plane with uniform distortion.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"128 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870121","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 M be a closed Riemannian manifold with Kazhdan fundamental group. It is well known that the Cheeger inequality yields a uniform waist inequality in codimension one for the finite covers of M. We show that the finite covers of M also satisfy a uniform waist inequality in codimension two.
设 M 是具有卡氏基本群的封闭黎曼流形。众所周知，Cheeger 不等式产生了 M 的有限盖在标度一中的均匀腰不等式。
{"title":"Uniform waist inequalities in codimension two for manifolds with Kazhdan fundamental group","authors":"Uri Bader, Roman Sauer","doi":"arxiv-2407.19783","DOIUrl":"https://doi.org/arxiv-2407.19783","url":null,"abstract":"Let M be a closed Riemannian manifold with Kazhdan fundamental group. It is\u0000well known that the Cheeger inequality yields a uniform waist inequality in\u0000codimension one for the finite covers of M. We show that the finite covers of M\u0000also satisfy a uniform waist inequality in codimension two.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870126","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}
The Minkowski problem in convex geometry concerns showing a given Borel measure on the unit sphere is, up to perhaps a constant, some type of surface area measure of a convex body. Two types of Minkowski problems in particular are an active area of research: $L^p$ Minkowski problems, introduced by Lutwak and (Lutwak,Yang, and Zhang), and weighted Minkowski problems, introduced by Livshyts. For the latter, the Gaussian Minkowski problem, whose primary investigators were (Huang, Xi and Zhao), is the most prevalent. In this work, we consider weighted surface area in the $L^p$ setting. We propose a framework going beyond the Gaussian setting by focusing on rotational invariant measures, mirroring the recent development of the Gardner-Zvavitch inequality for rotational invariant, log-concave measures. Our results include existence for all $p in mathbb R$ (with symmetry assumptions in certain instances). We also have uniqueness for $p geq 1$ under a concavity assumption. Finally, we obtain results in the so-called $small$ $mass$ $regime$ using degree theory, as instigated in the Gaussian case by (Huang, Xi and Zhao). Most known results for the Gaussian Minkowski problem are then special cases of our main theorems.
{"title":"The Weighted $L^p$ Minkowski Problem","authors":"Dylan Langharst, Jiaqian Liu, Shengyu Tang","doi":"arxiv-2407.20064","DOIUrl":"https://doi.org/arxiv-2407.20064","url":null,"abstract":"The Minkowski problem in convex geometry concerns showing a given Borel\u0000measure on the unit sphere is, up to perhaps a constant, some type of surface\u0000area measure of a convex body. Two types of Minkowski problems in particular\u0000are an active area of research: $L^p$ Minkowski problems, introduced by Lutwak\u0000and (Lutwak,Yang, and Zhang), and weighted Minkowski problems, introduced by\u0000Livshyts. For the latter, the Gaussian Minkowski problem, whose primary\u0000investigators were (Huang, Xi and Zhao), is the most prevalent. In this work,\u0000we consider weighted surface area in the $L^p$ setting. We propose a framework\u0000going beyond the Gaussian setting by focusing on rotational invariant measures,\u0000mirroring the recent development of the Gardner-Zvavitch inequality for\u0000rotational invariant, log-concave measures. Our results include existence for\u0000all $p in mathbb R$ (with symmetry assumptions in certain instances). We also\u0000have uniqueness for $p geq 1$ under a concavity assumption. Finally, we obtain\u0000results in the so-called $small$ $mass$ $regime$ using degree theory, as\u0000instigated in the Gaussian case by (Huang, Xi and Zhao). Most known results for\u0000the Gaussian Minkowski problem are then special cases of our main theorems.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870124","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}
The aim of this note is to survey the results in some geometric problems related to the centroids and the static equilibrium points of convex bodies. In particular, we collect results related to Gr"unbaum's inequality and the Busemann-Petty centroid inequality, describe classifications of convex bodies based on equilibrium points, and investigate the location and structure of equilibrium points, their number with respect to a general reference point as well as the static equilibrium properties of convex polyhedra.
{"title":"Centroids and equilibrium points of convex bodies","authors":"Zsolt Lángi, Péter L. Várkonyi","doi":"arxiv-2407.19177","DOIUrl":"https://doi.org/arxiv-2407.19177","url":null,"abstract":"The aim of this note is to survey the results in some geometric problems\u0000related to the centroids and the static equilibrium points of convex bodies. In\u0000particular, we collect results related to Gr\"unbaum's inequality and the\u0000Busemann-Petty centroid inequality, describe classifications of convex bodies\u0000based on equilibrium points, and investigate the location and structure of\u0000equilibrium points, their number with respect to a general reference point as\u0000well as the static equilibrium properties of convex polyhedra.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"213 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870123","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 the Morse boundary of a Morse local-to-global group is $sigma$-compact. Moreover, we show that the converse holds for small cancellation groups. As an application, we show that the Morse boundary of a non-hyperbolic, Morse local-to-global group that has contraction does not admit a non-trivial stationary measure. In fact, we show that any stationary measure on a geodesic boundary of such a groups needs to assign measure zero to the Morse boundary. Unlike previous results, we do not need any assumptions on the stationary measures considered.
{"title":"Sigma-compactness of Morse boundaries in Morse local-to-global groups and applications to stationary measures","authors":"Vivian He, Davide Spriano, Stefanie Zbinden","doi":"arxiv-2407.18863","DOIUrl":"https://doi.org/arxiv-2407.18863","url":null,"abstract":"We show that the Morse boundary of a Morse local-to-global group is\u0000$sigma$-compact. Moreover, we show that the converse holds for small\u0000cancellation groups. As an application, we show that the Morse boundary of a\u0000non-hyperbolic, Morse local-to-global group that has contraction does not admit\u0000a non-trivial stationary measure. In fact, we show that any stationary measure\u0000on a geodesic boundary of such a groups needs to assign measure zero to the\u0000Morse boundary. Unlike previous results, we do not need any assumptions on the\u0000stationary measures considered.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870129","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, inside the Shilov boundary of any given Hermitian symmetric space of tube type, there is, up to isomorphism, only one proper domain whose action by its automorphism group is cocompact. This gives a classification of all closed proper manifolds locally modelled on such Shilov boundaries, and provides a positive answer, in the case of flag manifolds admitting a $Theta$-positive structure, to a rigidity question of Limbeek and Zimmer.
{"title":"Rigidity of proper quasi-homogeneous domains in positive flag manifolds","authors":"Blandine Galiay","doi":"arxiv-2407.18747","DOIUrl":"https://doi.org/arxiv-2407.18747","url":null,"abstract":"We show that, inside the Shilov boundary of any given Hermitian symmetric\u0000space of tube type, there is, up to isomorphism, only one proper domain whose\u0000action by its automorphism group is cocompact. This gives a classification of\u0000all closed proper manifolds locally modelled on such Shilov boundaries, and\u0000provides a positive answer, in the case of flag manifolds admitting a\u0000$Theta$-positive structure, to a rigidity question of Limbeek and Zimmer.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"106 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870130","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}
The Einstein universe $mathbf{Ein}^{p,q}$ of signature $(p,q)$ is a pseudo-Riemannian analogue of the conformal sphere; it is the conformal compactification of the pseudo-Riemannian Minkowski space. For $p,q geq 1$, we show that, up to a conformal transformation, there is only one domain in $mathbf{Ein}^{p,q}$ that is bounded in a suitable stereographic projection and whose action by its conformal group is cocompact. This domain, which we call a diamond, is a model for the symmetric space of $operatorname{PO}(p,1) times operatorname{PO}(1,q)$. We deduce a classification of closed conformally flat manifolds with proper development.
{"title":"Proper quasi-homogeneous domains of the Einstein universe","authors":"Adam ChalumeauIRMA, Blandine GaliayIHES","doi":"arxiv-2407.18577","DOIUrl":"https://doi.org/arxiv-2407.18577","url":null,"abstract":"The Einstein universe $mathbf{Ein}^{p,q}$ of signature $(p,q)$ is a\u0000pseudo-Riemannian analogue of the conformal sphere; it is the conformal\u0000compactification of the pseudo-Riemannian Minkowski space. For $p,q geq 1$, we\u0000show that, up to a conformal transformation, there is only one domain in\u0000$mathbf{Ein}^{p,q}$ that is bounded in a suitable stereographic projection and\u0000whose action by its conformal group is cocompact. This domain, which we call a\u0000diamond, is a model for the symmetric space of $operatorname{PO}(p,1) times\u0000operatorname{PO}(1,q)$. We deduce a classification of closed conformally flat\u0000manifolds with proper development.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"295 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870131","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}
Borell's inequality states the existence of a positive absolute constant $C>0$ such that for every $1leq pleq q$ $$ left(mathbb E|langle X, e_nrangle|^pright)^frac{1}{p}leqleft(mathbb E|langle X, e_nrangle|^qright)^frac{1}{q}leq Cfrac{q}{p}left(mathbb E|langle X, e_nrangle|^pright)^frac{1}{p}, $$ whenever $X$ is a random vector uniformly distributed in any convex body $Ksubseteqmathbb R^n$ containing the origin in its interior and $(e_i)_{i=1}^n$ is the standard canonical basis in $mathbb R^n$. In this paper, we will prove a discrete version of this inequality, which will hold whenever $X$ is a random vector uniformly distributed on $Kcapmathbb Z^n$ for any convex body $Ksubseteqmathbb R^n$ containing the origin in its interior. We will also make use of such discrete version to obtain discrete inequalities from which we can recover the estimate $mathbb E w(K_N)sim w(Z_{log N}(K))$ for any convex body $K$ containing the origin in its interior, where $K_N$ is the centrally symmetric random polytope $K_N=operatorname{conv}{pm X_1,ldots,pm X_N}$ generated by independent random vectors uniformly distributed on $K$ and $w(cdot)$ denotes the mean width.
{"title":"Borell's inequality and mean width of random polytopes via discrete inequalities","authors":"David Alonso-Gutiérrez, Luis C. García-Lirola","doi":"arxiv-2407.18235","DOIUrl":"https://doi.org/arxiv-2407.18235","url":null,"abstract":"Borell's inequality states the existence of a positive absolute constant\u0000$C>0$ such that for every $1leq pleq q$ $$ left(mathbb E|langle X,\u0000e_nrangle|^pright)^frac{1}{p}leqleft(mathbb E|langle X,\u0000e_nrangle|^qright)^frac{1}{q}leq Cfrac{q}{p}left(mathbb E|langle X,\u0000e_nrangle|^pright)^frac{1}{p}, $$ whenever $X$ is a random vector uniformly\u0000distributed in any convex body $Ksubseteqmathbb R^n$ containing the origin in\u0000its interior and $(e_i)_{i=1}^n$ is the standard canonical basis in $mathbb\u0000R^n$. In this paper, we will prove a discrete version of this inequality, which\u0000will hold whenever $X$ is a random vector uniformly distributed on\u0000$Kcapmathbb Z^n$ for any convex body $Ksubseteqmathbb R^n$ containing the\u0000origin in its interior. We will also make use of such discrete version to\u0000obtain discrete inequalities from which we can recover the estimate $mathbb E\u0000w(K_N)sim w(Z_{log N}(K))$ for any convex body $K$ containing the origin in\u0000its interior, where $K_N$ is the centrally symmetric random polytope\u0000$K_N=operatorname{conv}{pm X_1,ldots,pm X_N}$ generated by independent\u0000random vectors uniformly distributed on $K$ and $w(cdot)$ denotes the mean\u0000width.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785618","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}
Ryan Gibara, Ilmari Kangasniemi, Nageswari Shanmugalingam
We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure $mu$ with $mu(X) = infty$ and $0 < mu(B(x, r)) < infty$ for all $x in X$ and $r in (0, infty)$ Our objective is to understand the relationship between the Dirichlet space $D^{1,p}(X)$, defined using upper gradients, and the Newton-Sobolev space $N^{1,p}(X)+mathbb{R}$, for $1le p
{"title":"On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry","authors":"Ryan Gibara, Ilmari Kangasniemi, Nageswari Shanmugalingam","doi":"arxiv-2407.18315","DOIUrl":"https://doi.org/arxiv-2407.18315","url":null,"abstract":"We study the large-scale behavior of Newton-Sobolev functions on complete,\u0000connected, proper, separable metric measure spaces equipped with a Borel\u0000measure $mu$ with $mu(X) = infty$ and $0 < mu(B(x, r)) < infty$ for all $x\u0000in X$ and $r in (0, infty)$ Our objective is to understand the relationship\u0000between the Dirichlet space $D^{1,p}(X)$, defined using upper gradients, and\u0000the Newton-Sobolev space $N^{1,p}(X)+mathbb{R}$, for $1le p<infty$. We show\u0000that when $X$ is of uniformly locally $p$-controlled geometry, these two spaces\u0000do not coincide under a wide variety of geometric and potential theoretic\u0000conditions. We also show that when the metric measure space is the standard\u0000hyperbolic space $mathbb{H}^n$ with $nge 2$, these two spaces coincide\u0000precisely when $1le ple n-1$. We also provide additional characterizations of\u0000when a function in $D^{1,p}(X)$ is in $N^{1,p}(X)+mathbb{R}$ in the case that\u0000the two spaces do not coincide.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870128","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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