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

arXiv: https://arxiv.org/abs/2107.02223 · HTML: https://mlsanity.com/paper/2107.02223/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jul 2021.

## Built on by (most cited first, 2 of 2)

- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)

## Abstract (arXiv)

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.