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In this note, we show that the squared Wasserstein distance can be expressed as the average over the sphere of one dimensional Wasserstein distances. We name this new expression for the Wasserstein Distance Radiant Formula. Using this formula, we are able to highlight new connections between the Wasserstein distances and the Sliced Wasserstein distance, an alternative Wasserstein-like distance that is cheaper to compute. [ABSTRACT FROM AUTHOR]

The local slope operator was introduced in De Giorgi et al. (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Nat 68:180–187, 1980) to study gradient flow dynamics in metric spaces. This tool has now become a cornerstone in metric evolution equations (see, e.g., Ambrosio et al. (Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics. Birkhauser, 2008). Very recently, in Daniilidis and Salas (Proc Am Math Soc 150:4325–4333, 2022), it was established that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, ensuring in particular the existence of critical points. We hereby emancipate from this restriction and establish a determination result for merely bounded from below functions, by adding an assumption controlling the asymptotic behavior. This assumption is trivially fulfilled if f is inf-compact. In addition, our result is not only valid for the (De Giorgi) local slope, but also for the main paradigms of average descent operators as well as for the global slope, case in which the asymptotic assumption becomes superfluous. Therefore, the present work extends simultaneously the metric determination results of Daniilidis and Salas (Proc Am Math Soc 150:4325–4333, 2022) and Thibault and Zagrodny (Commun Contemp Math 25(7):2250014, 2023). [ABSTRACT FROM AUTHOR]

We study the reach (in the sense of Federer) of the natural isometric embedding X ↪ W p (X) of X inside its p-Wasserstein space, where (X , dist) is a geodesic metric space. We prove that if a point x ∈ X can be joined to another point y ∈ X by two minimizing geodesics, then reach (x , X ⊂ W p (X)) = 0 . This includes the cases where X is a compact manifold or a non-simply connected one. On the other hand, we show that reach (X ⊂ W p (X)) = ∞ when X is a CAT(0) space. The infinite reach enables us to examine the regularity of the projection map. Furthermore, we replicate these findings by considering the isometric embedding X ↪ W ϑ (X) into an Orlicz–Wasserstein space, a generalization by Sturm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of X ↪ Dgm ∞ , the space of persistence diagrams equipped with the bottleneck distance. [ABSTRACT FROM AUTHOR]

This paper is motivated by the classical theorem due to Hardy and Littlewood which concerns analytic mappings on the unit disk and relates the growth of the derivative with the Hölder continuity. We obtain a version of this result in a very general setting—for regularly oscillating mappings on a metric space equipped with a weight, which is a continuous and positive function, with values in another metric space. As a consequence, we derive the Hardy and Littlewood theorem for analytic mappings on the unit ball of a normed space. [ABSTRACT FROM AUTHOR]

We introduce a natural notion of mean (or average) distance in the context of compact metric graphs, and study its relation to geometric properties of the graph. We show that it exhibits a striking number of parallels to the reciprocal of the spectral gap of the graph Laplacian with standard vertex conditions: it is maximised among all graphs of fixed length by the path graph (interval), or by the loop in the restricted class of doubly connected graphs, and it is minimised among all graphs of fixed length and number of edges by the equilateral flower graph. We also establish bounds for the correctly scaled product of the spectral gap and the square of the mean distance which depend only on combinatorial, and not metric, features of the graph. This raises the open question whether this product admits absolute upper and lower bounds valid on all compact metric graphs. [ABSTRACT FROM AUTHOR]

Every graph of bounded degree endowed with the counting measure satisfies a local version of Lp-Poincaré inequality, p ∈ [1, ∞]. We show that on graphs which are trees the Poincaré constant grows at least exponentially with the radius of balls. On the other hand, we prove that, surprisingly, trees endowed with a flow measure support a global version of Lp-Poincaré inequality, despite the fact that they are nondoubling measures of exponential growth. [ABSTRACT FROM AUTHOR]

In this manuscript, some remarks on the papers [H. A. Hammad, P. Agarwal, S. Momani, and F. Alsharari, Solving a fractional-order differential equation using rational symmetric contraction mappings, Fractal Fract. 5 (2021), 159] and [A. Hussain, F. Jarad, and E. Karapinar, A study of symmetric contractions with an application to generalized fractional differential equations, Adv. Differ. Equ. 2021 (2021), 300] are given. In the light of remarks, we introduce a new property that makes it convenient to investigate the existence of fixed points of the interpolative contractions in the orthogonal metric spaces. We derive several new results based on known contractions from the main theorems. As an application, we resolve a Urysohn integral equation.

We give a transference maximum inequality establishing interrelation between the sign-convergence and the rearrangement convergence of functional series. We use this inequality to get a generalization of the Maurey and Pisier sign-permutation relationship and the general Garsia and Nikishin type local inequalities, as well as the Nikishin and Garsia convergence theorems on the almost sure convergence of functional series. We also get some maximum inequalities that might be useful in the analysis of the Kolmogorov Rearrangement Conjecture on the systems of convergence of orthogonal series as well as Garsia's local form of the conjecture. [ABSTRACT FROM AUTHOR]

We give another proof of Toyoda’s theorem that describes 5-point subspaces in CAT(0) length spaces.

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipschitz clustering property, while for some connected spaces, the lack of sufficiently short connecting curves turns out to be an obstruction. This property is shown to be invariant under quasihomogeneous maps, but not under quasisymmetric ones. [ABSTRACT FROM AUTHOR]

In Ketterer and Rajala (Potential Anal 42:645–655, 2014) showed an example of metric measure space, satisfying the measure contraction property MCP (0 , 3) , that has different topological dimensions at different regions of the space. In this article I propose a refinement of that example, which satisfies the CD (0 , ∞) condition, proving the non-constancy of topological dimension for CD spaces. This example also shows that the weak curvature dimension bound, in the sense of Lott–Sturm–Villani, is not sufficient to deduce any reasonable non-branching condition. Moreover, it allows to answer to some open question proposed by Schultz in (Calc Var Partial Differ Equ 57:1–11, 2018), about strict curvature dimension bounds and their stability with respect to the measured Gromov–Hausdorff convergence. [ABSTRACT FROM AUTHOR]