A Takahashi convex structure is considered on Menger PM-spaces and used to investigate the existence of best proximity points for weak cyclic Kannan contractions. We then introduce a concept of a probabilistic proximal quasi-normal structure on a convex pair of subsets of Menger PM-spaces and prove that every compact and convex pair in metrically convex Menger PM-spaces has the probabilistic proximal quasi-normal structure. By applying this geometric property, we survey the existence of a best proximity point for cyclic relatively Kannan nonexpansive maps which preserves distance. In order to provide more accurate results, we obtain the same conclusions in the framework of CAT(0) spaces.
Gabeleh et al. (2025) studied this question.