Geodesic
Axiomatic theory of convexity
Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 761-770.

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The axiomatic construction of the theory of convexity proceeds from an arbitrary set $M$ and a mapping $l:M^2\to2^M$, i.e., from a pair $(M,l)$. It is shown that such a space of a certain type is domain finite. A condition is given which, for such spaces, implies join-hull commutativity. A connection is established between the Carathéodory number and join-hull commutativity. Conditions are given which imply a separation property of the space $(M,l)$. Convexity spaces which are domain finite are characterized. 
@article{MZM_1976_20_5_a16,
     author = {V. V. Tuz},
     title = {Axiomatic theory of convexity},
     journal = {Matemati\v{c}eskie zametki},
     pages = {761--770},
     publisher = {mathdoc},
     volume = {20},
     number = {5},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a16/}
}
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V. V. Tuz. Axiomatic theory of convexity. Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 761-770. http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a16/