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In this article, we introduce a geometrical notion, property strongly UCwhich is stronger than property UC and prove the existence of bestapproximations for a new class of almost cyclic $\psi$-contraction maps in ametric space. As a particular case, we obtain the main results of [Sadiq Basha,S., Best approximation theorems for almost cyclic contractions. J. Fixed PointTheory Appl. 23 (2021), MR4254305] and [Eldred, A. Anthony; Veeramani, P.,Existence and convergence of best proximity points. J. Math. Anal. Appl. 323(2006), MR2260159]. Moreover, we study the existence of best approximations foralmost cyclic contractions in a reflexive Banach space.

Best approximations in metric spaces with property strongly UC