  In this article, we introduce a geometrical notion, property strongly UC
which is stronger than property UC and prove the existence of best
approximations for a new class of almost cyclic $\psi$-contraction maps in a
metric space. As a particular case, we obtain the main results of [Sadiq Basha,
S., Best approximation theorems for almost cyclic contractions. J. Fixed Point
Theory 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 for
almost cyclic contractions in a reflexive Banach space.
