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1. Topological k-Metrics

   https://doi.org/10.4230/LIPIcs.SoCG.2024.13  

   Barkan, Willow; Bennett, Huck; Nayyeri, Amir (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Metric spaces (X, d) are ubiquitous objects in mathematics and computer science that allow for capturing pairwise distance relationships d(x, y) between points x, y ∈ X. Because of this, it is natural to ask what useful generalizations there are of metric spaces for capturing "k-wise distance relationships" d(x_1, …, x_k) among points x_1, …, x_k ∈ X for k > 2. To that end, Gähler (Math. Nachr., 1963) (and perhaps others even earlier) defined k-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality d(x₁, x₂) ≤ d(x₁, y) + d(y, x₂) to the "simplex inequality" d(x_1, …, x_k) ≤ ∑_{i=1}^k d(x_1, …, x_{i-1}, y, x_{i+1}, …, x_k). (The definition holds for any fixed k ≥ 2, and a 2-metric space is just a (standard) metric space.) In this work, we introduce strong k-metric spaces, k-metric spaces that satisfy a topological condition stronger than the simplex inequality, which makes them "behave nicely." We also introduce coboundary k-metrics, which generalize 𝓁_p metrics (and in fact all finite metric spaces induced by norms) and minimum bounding chain k-metrics, which generalize shortest path metrics (and capture all strong k-metrics). Using these definitions, we prove analogs of a number of fundamental results about embedding finite metric spaces including Fréchet embedding (isometric embedding into 𝓁_∞) and isometric embedding of all tree metrics into 𝓁₁. We also study relationships between families of (strong) k-metrics, and show that natural quantities, like simplex volume, are strong k-metrics. 

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2. Classification of metric measure spaces and their ends using p-harmonic functions

   https://doi.org/10.54330/afm.120618  

   Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (March 2022, Annales Fennici Mathematici)

   By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite \(p\)-energy \(p\)-harmonic and \(p\)-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local \(p\)-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the \(p\)-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant \(p\)-harmonic functions with finite \(p\)-energy as spaces having at least two well-separated \(p\)-hyperbolic sequences of sets towards infinity. We also show that every such space \(X\) has a function \(f \notin L^p(X) + \mathbf{R}\) with finite \(p\)-energy. 

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3. On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry

   https://doi.org/10.1007/s00229-025-01656-5  

   Gibara, Ryan; Kangasniemi, Ilmari; Shanmugalingam, Nageswari (July 2025, Manuscripta mathematica)

   We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure μ with μ(X) = ∞ and 0 < μ(B(x, r)) < ∞ for all x ∈ X and r ∈ (0, ∞). 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)+ℝ, for 1 ≤ p < ∞. We show that when X is of uniformly locally p-controlled geometry, these two spaces do not coincide under a wide variety of geometric and potential theoretic conditions. We also show that when the metric measure space is the standard hyperbolic space ℍⁿ with n ≥ 2, these two spaces coincide precisely when 1 ≤ p ≤ n-1. We also provide additional characterizations of when a function in D^(1,p)(X) is in N^(1,p)(X)+ℝ in the case that the two spaces do not coincide. 

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4. Design Variety Measurement using Sharma-Mittal Entropy

   https://doi.org/10.1115/1.4048743  

   Ahmed, Faez; Ramachandran, Sharath Kumar; Fuge, Mark; Hunter, Samuel; Miller, Scarlett (January 2020, Journal of Mechanical Design)

   null (Ed.)

   Design variety metrics measure how much a design space is explored. We propose that a generalized class of entropy measures based on Sharma-Mittal entropy offers advantages over existing methods to measure design variety. We show that an exemplar metric from Sharma-Mittal entropy, which we call the Herfindahl–Hirschman Index for Design (HHID) has the following desirable advantages over existing metrics: (a) More Accuracy: It better aligns with human ratings compared to existing and commonly used tree-based metrics for two new datasets; (b) Higher Sensitivity: It has higher sensitivity compared to existing methods when distinguishing between the variety of sets; (c) Allows Efficient Optimization: It is a submodular function, which enables us to optimize design variety using a polynomial-time greedy algorithm; and (d) Generalizes to Multiple Measures: The parametric nature of this metric allows us to fit the metric to better represent variety for new domains. The paper also contributes a procedure for comparing metrics used to measure variety via constructing ground truth datasets from pairwise comparisons. Overall, our results shed light on some qualities that good design variety metrics should possess and the non-trivial challenges associated with collecting the data needed to measure those qualities. 

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5. Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces

   https://doi.org/10.1090/tran/8495  

   Eriksson-Bique, Sylvester; Gill, James T.; Lahti, Panu; Shanmugalingam, Nageswari (November 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincare inequality. We show that at almost every point x outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at x. We also show that, at co-dimension 1 Hausdorff measure almost every measure-theoretic boundary point of a set E of finite perimeter, there is an asymptotic limit set (E)∞ corresponding to the asymptotic expansion of E and that every such asymptotic limit (E)∞ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of (E)∞ is Ahlfors co-dimension 1 regular. 

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