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In this article, we formulate a best proximity pair theorem for noncyclic relatively nonexpansive mappings in convex metrc spaces by using a geometric notion of semi-normal structure. In this way, we generalize a corresponding result in [W. Takahashi, A convexity in metric space and nonexpansive mappings, KODAI MATH. SEM. REP. 22 (1970) 142-149]. We also establish a best proximity pair theorem for pointwise noncyclic contractions in the setting of convex metric spaces. Our result generalizes a result due to Sankara Raju Kosuru and Veeramani [G. Sankara Raju Kosuru and P. Veeramani, A note on existence and convergence of best proximity points for pointwise cyclic contractions, Numer. Funct. Anal. Optim., 82 (2011) 821-830].

Best proximity pair and fixed point results for noncyclic mappings in convex metric spaces