Aug 2026· Israel Journal of Mathematics· 4 citations· 27 references
Abstract
For general hyperbolic metric spaces, we introduce a new notion of a dual system (extending the influential notion from the context of normed linear spaces) that allows for a uniform study of different notions of duality for these nonlinear spaces. Using this abstract notion of duality, we lift various notions from convex analysis into this nonlinear setting, including Fréchet differentiability and Bregman distances. Further, we introduce a notion of a monotone operator relative to a given dual system and, using the new Fréchet derivatives, we study corresponding resolvents relative to a given gradient, generalizing the seminal notion of Eckstein from the linear setting. These resolvents are then related to corresponding notions of Bregman nonexpansive mappings which are introduced relative to this generalization of the classical Bregman distance and we prove a convergence result of an analogue of the proximal point algorithm. For that, using methods from proof mining, we even provide quantitative results on its convergence in very general settings.
Using a novel effective non-asymptotic concentration inequality, we establish a distribution-uniform generalization of Sturm's strong law of large numbers for inductive means of $L^1$-sequences of i.i.d. random variables on (separable) Hadamard spaces. Building on that, we establish a strong law of large numbers for in...
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et...
T. Hytönen, Da-Chun Yang, Wen Yuan et al.· 2 citations
Considering an $n$-dimensional compact Riemannian manifold with a boundary that satisfies a Serrin-type problem, we prove sharp upper and lower bounds for the area of such a boundary. Then, we present the main result of this paper: a divergence formula for a special vector field on a given Riemannian manifold. This div...
D. Yost [ Intersecting balls and proximinal subspaces in Banach spaces, Ph.D. Thesis, University of Edinburgh (1979)] introduced a notion called equability for subspaces of a Banach space, as a sufficient condition for the proximinality and lower semi-continuity of the metric projection. This notion was later modifie...
P. Gayathri, K. Sree Lakshmi, V. Thota· Journal of Convex Analysis· 0 citations
In this article we consider various properties of a normed lattice, which are similar to semicontinuity (also known as the Fatou property), establish relations between these properties, and discuss stability of these properties under renorming. In particular, we show that a normed lattice $F$ is order continuous iff ev...
The problem of differentiating Lipschitz curves in Banach spaces goes back to Tamarkin in the early 1930s and is one of the origins of the Radon--Nikod\'ym property in classical Banach space theory. In this article we characterize those quasi-Banach spaces $X$ for which every Lipschitz curve $F\to\mathbb{R}\to X$ admit...
F. Albiac, J. Ansorena, Pietro Wald· 1 citation
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