Convexity and Hausdorff-Pompeiu Distance
Mathematica Moravica, Tome 15 (2011) no. 1.

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The aim of this paper is to realize a decomposition of the usual convexity structures on metric spaces. Thus, a metric space is totally convex if and only if it satisfies the conditions (A) and (B) (Proposition 2). Also, it is totally externally convex if and only if both conditions (A') and (B') are satisfied (Proposition 4). Some connections between the convexity conditions (A) and (A') and the Hausdorff-Pompeiu metric are investigated (see, for example, Corollary 3).
Mots-clés : Convexity condition (A), convexity condition (B), totally convex, Hausdorff-Pompeiu metric 
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Temistocle Bîrsan. Convexity and Hausdorff-Pompeiu Distance. Mathematica Moravica, Tome 15 (2011) no. 1. https://geodesic-test.mathdoc.fr/item/MM3_2011_15_1_a2/