arXiv subjects

Francisco Venegas M

Publications and source records attributed to Francisco Venegas M.

4 recordsLinked to original sources

arXiv2026

The reach and limits of slope eikonal equations in compact spaces

It is a well known fact that the eikonal equation is well posed in complete length spaces. Among the studied notions of solutions in the literature, there is one that can be defined in any metric space using the local (descent) slope and considering pointwise solutions: functionals such that their slope coincides with the prescribed data at every point of the domain. In this work we explore the question ``Can we characterize the class of compact metric spaces in which every slope eikonal equation (under standard assumptions) always admits a pointwise solution?''. We provide a purely metric characterization of these spaces, as well as some interesting examples and counterexamples that illustrate the reach and limitations of the concept.

math.FA↗

arXiv2025

A functional characterization of isometries between non-reversible Finsler manifolds

We provide a functional characterization of isometries between non-reversible Finsler manifolds, in the form of a generalization of the Myers-Nakai Theorem for Riemannian manifolds. We show that, since non-reversible Finsler manifolds are a fundamentally asymmetric object, such a result can not be obtained by means of a symmetric function space, and we define the appropriate asymmetric structure needed to describe all possible isometries between this class of manifolds. The result is based on the ideas used in a previous generalization for reversible Finsler manifolds proved in \cite{GJR-13}, in which the normed algebra of $C^1$-smooth Lipschitz functions is used. To reflect the quasi-metric structure of non-reversible Finsler manifolds, this normed algebra had to be modified to include the cone of smooth semi-Lipschitz functions, resulting in a partial loss of the normed space and algebra structures. In order to achieve the desired result, we define new algebraic/quasi-metric structures to model the behavior of the aforementioned function space.

math.FA↗

arXiv2024

Pointwise semi-Lipschitz functions and Banach-Stone theorems

We study the fundamental properties of pointwise semi-Lipschitz functions between asymmetric spaces, which are the natural asymmetric counterpart of pointwise Lipschitz functions. We also study the influence that partial symmetries of a given space may have on the behavior of pointwise semi-Lipschitz functions defined on it. Furthermore, we are interested in characterizing the pointwise semi-Lipschitz structure of an asymmetric space in terms of real-valued pointwise semi-Lipschitz functions defined on it. By using two algebras of functions naturally associated to our spaces of pointwise real-valued semi-Lipschitz functions, we are able to provide two Banach-Stone type results in this context. In fact, these results are obtained as consequences of a general Banach-Stone type theorem of topological nature, stated for abstract functional spaces, which is quite flexible and can be applied to many spaces of continuous functions over metric and asymmetric spaces.

math.FA↗

arXiv2019

Smooth semi-Lipschitz functions and almost isometries of Finsler manifolds

The convex cone $SC_{\mathrm{SLip}}^1(\mathcal{X})$ of real-valued smooth semi-Lipschitz functions on a Finsler manifold $\mathcal{X}$ is an order-algebraic structure that captures both the differentiable and the quasi-metric feature of $\mathcal{X}$. In this work we show that the subset of smooth semi-Lipschitz functions of constant strictly less than $1$, denoted $SC_{1^{-}}^1(\mathcal{X})$, can be used to classify Finsler manifolds and to characterize almost isometries between them, in the lines of the classical Banach-Stone and Mykers-Nakai theorems.

math.FA↗