Elements of Convex Geometry in Hadamard Manifolds with Application to Equilibrium Problems

TL;DR

This paper introduces a new resolvent for equilibrium problems on Hadamard manifolds using Busemann's function, extending linear case methods and exploring convex analysis in non-linear geometric settings.

Contribution

It proposes a convex regularization term in Hadamard manifolds and develops convex analysis tools like a new convex combination and Jensen inequality.

Findings

New resolvent based on Busemann's function for equilibrium problems

Convex regularization term is applicable in general Hadamard manifolds

Established a Jensen-type inequality in the manifold setting

Abstract

In this paper, is introduced a new proposal of resolvent for equilibrium problems in terms of the Busemann's function. A great advantage of this new proposal is that, in addition to be a natural extension of the proposal in the linear setting by Combettes and Hirstoaga in [20], the new term that performs regularization is a convex function in general Hadamard manifolds, being a first step to fully answer to the problem posed by Cruz Neto et al. in [21, Section 5]. During our study, some elements of convex analysis are explored in the context of Hadamard manifolds, which are interesting on their own. In particular, we introduce a new definition of convex combination (now commutative) of any finite collection of points and present the realization of an associated Jensen-type inequality.