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In this paper, is introduced a new proposal of resolvent for equilibriumproblems in terms of the Busemann's function. A great advantage of this newproposal is that, in addition to be a natural extension of the proposal in thelinear setting by Combettes and Hirstoaga in [20], the new term that performsregularization is a convex function in general Hadamard manifolds, being afirst 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 inthe context of Hadamard manifolds, which are interesting on their own. Inparticular, we introduce a new definition of convex combination (nowcommutative) of any finite collection of points and present the realization ofan associated Jensen-type inequality.

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