Metric Geometry

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New submissions for Tue, 14 May 24

[1]  arXiv:2405.07329 [pdf, ps, other]

Title: New approach to affine Moser-Trudinger inequalities via Besov polar projection bodies

Authors: Oscar Dominguez, Yinqin Li, Sergey Tikhonov, Dachun Yang, Wen Yuan

Subjects: Metric Geometry (math.MG); Functional Analysis (math.FA)

We extend the affine inequalities on $\mathbb{R}^n$ for Sobolev functions in $W^{s,p}$ with $1 \leq p < n/s$ obtained recently by Haddad-Ludwig [16, 17] to the remaining range $p \geq n/s$. For each value of $s$, our results are stronger than affine Moser-Trudinger and Morrey inequalities. As a byproduct, we establish the analog of the classical $L^p$ Bourgain-Brezis-Mironescu inequalities related to the Moser-Trudinger case $p=n$. Our main tool is the affine invariant provided by Besov polar projection bodies.

[2]  arXiv:2405.07476 [pdf, ps, other]

Title: Definitions of quasiconformality on metric surfaces

Authors: Damaris Meier, Kai Rajala

Subjects: Metric Geometry (math.MG)

We explore the interplay between different definitions of distortion for mappings $f\colon X\to \mathbb{R}^2$, where $X$ is any metric surface, meaning that $X$ is homeomorphic to a domain in $\mathbb{R}^2$ and has locally finite 2-dimensional Hausdorff measure. We establish that finite distortion in terms of the familiar analytic definition always implies finite distortion in terms of maximal and minimal stretchings along paths. The converse holds for maps with locally integrable distortion. In particular, we prove the equivalence of various notions of quasiconformality, implying a novel uniformization result for metric surfaces.

Cross-lists for Tue, 14 May 24

[3]  arXiv:2405.07253 (cross-list from math.PR) [pdf, ps, other]

Title: Sharp estimates for the Cramér transform of log-concave measures and geometric applications

Authors: Silouanos Brazitikos, Giorgos Chasapis

Subjects: Probability (math.PR); Functional Analysis (math.FA); Metric Geometry (math.MG)

We establish a new comparison between the Legendre transform of the cumulant generating function and the half-space depth of an arbitrary log-concave probability distribution on the real line, that carries on to the multidimensional setting. Combined with sharp estimates for the Cram\'{e}r transform of rotationally invariant measures, we are led to some new phase-transition type results for the asymptotics of the expected measure of random polytopes. As a byproduct of our analysis, we address a question on the sharp exponential separability constant for log-concave distributions, in the symmetric case.

[4]  arXiv:2405.07512 (cross-list from math.CO) [pdf, ps, other]

Title: Separation axiom $S_3$ for geodesic convexity in graphs

Authors: Victor Chepoi

Comments: 59 pages, 2 figures

Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Metric Geometry (math.MG)

Semispaces of a convexity space $(X,C)$ are maximal convex sets missing a point. The separation axiom $S_3$ asserts that any point $x_0\in X$ and any convex set $A$ not containing $x_0$ can be separated by complementary halfspaces (convex sets with convex complements) or, equivalently, that all semispaces are halfspaces. In this paper, we study $S_3$ for geodesic convexity in graphs and the structure of semispaces in $S_3$-graphs. We characterize $S_3$-graphs and their semispaces in terms of separation by halfspaces of vertices $x_0$ and special sets, called maximal $x_0$-proximal sets and in terms of convexity of their mutual shadows $x_0/K$ and $K/x_0$. In $S_3$-graphs $G$ satisfying the triangle condition (TC), maximal proximal sets are the pre-maximal cliques of $G$ (i.e., cliques $K$ such that $K\cup\{ x_0\}$ are maximal cliques). This allows to characterize the $S_3$-graphs satisfying (TC) in a structural way and to enumerate their semispaces efficiently. In case of meshed graphs (an important subclass of graphs satisfying (TC)), the $S_3$-graphs have been characterized by excluding five forbidden subgraphs. On the way of proving this result, we also establish some properties of meshed graphs, which maybe of independent interest. In particular, we show that any connected, locally-convex set of a meshed graph is convex. We also provide several examples of $S_3$-graphs, including the basis graphs of matroids. Finally, we consider the (NP-complete) halfspace separation problem, describe two methods of its solution, and apply them to particular classes of graphs and graph-convexities.

[5]  arXiv:2405.07688 (cross-list from math.GR) [pdf, ps, other]

Title: Finitely generated groups and harmonic functions of slow growth

Authors: Mayukh Mukherjee, Soumyadeb Samanta, Soumyadip Thandar

Comments: 20 pages, comments most welcome! arXiv admin note: text overlap with arXiv:1505.01175 by other authors

Subjects: Group Theory (math.GR); Metric Geometry (math.MG); Probability (math.PR)

In this paper, we are mainly concerned with $(\mathbb{G},\mu)$-harmonic functions that grow at most polynomially, where $\mathbb{G}$ is a finitely generated group with a probability measure $\mu$. In the initial part of the paper, we focus on Lipschitz harmonic functions and how they descend onto finite index subgroups. We discuss the relations between Lipschitz harmonic functions and harmonic functions of linear growth and conclude that for groups of polynomial growth, they coincide. In the latter part of the paper, we specialise to positive harmonic functions and give a characterisation for strong Liouville property in terms of the Green's function. We show that the existence of a non-constant positive harmonic function of polynomial growth guarantees that the group cannot have polynomial growth.

[6]  arXiv:2405.07818 (cross-list from math.CO) [pdf, ps, other]

Title: New lower bound on ball packing density in high-dimensional hyperbolic spaces

Authors: Irene Gil Fernández, Jaehoon Kim, Hong Liu, Oleg Pikhurko

Comments: 18 pages

Subjects: Combinatorics (math.CO); Metric Geometry (math.MG)

We present a new lower bound on the Bowen-Radin maximal density of radius-R ball packings in the m-dimensional hyperbolic space, improving on the basic covering bound by factor \Omega(m(R+\ln m)) as m tends to infinity. This is done by applying the recent theorem of Campos, Jenssen, Michelen and Sahasrabudhe on independent sets in graphs with sparse neighbourhoods.

Replacements for Tue, 14 May 24

[7]  arXiv:2303.16896 (replaced) [pdf, ps, other]

Title: Stability of polydisc slicing

Authors: Nathaniel Glover, Tomasz Tkocz, Katarzyna Wyczesany

Comments: Final version, presentation improved. To appear in Mathematika

Subjects: Metric Geometry (math.MG); Functional Analysis (math.FA); Probability (math.PR)

[8]  arXiv:2401.00786 (replaced) [pdf, ps, other]

Title: Magnitude function determines generic finite metric spaces

Authors: Jun O'Hara

Comments: 20 pages, 4 figures

Subjects: Metric Geometry (math.MG)

[9]  arXiv:2403.07900 (replaced) [pdf, ps, other]

Title: Applications of equidistant supporting surfaces of a convex body in the hyperbolic space

Authors: Marek Lassak

Comments: 7 pages. arXiv admin note: text overlap with arXiv:2401.07831

Subjects: Metric Geometry (math.MG)

[10]  arXiv:2111.05625 (replaced) [pdf, other]

Title: Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence

Authors: Amlan Banaji, István Kolossváry

Comments: v2: 50 pages, 5 figures. Added Theorem 2.4 and part (4) of Theorem 2.5. To appear in Advances in Mathematics

Subjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG)

[11]  arXiv:2401.07975 (replaced) [pdf, ps, other]

Title: Existence theorem for sub-Lorentzian problems

Authors: L.V. Lokutsievskiy, A.V. Podobryaev

Comments: 11 pages

Journal-ref: Journal of Dynamical and Control Systems. 30, 10 (2024)

Subjects: Differential Geometry (math.DG); Metric Geometry (math.MG); Optimization and Control (math.OC)

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