Geodesic
Partial convexity
Matematičeskie zametki, Tome 60 (1996) no. 3, pp. 406-413

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We consider a generalization of the classical notion of convexity, which is called partial convexity. Let $V\subseteq\mathbb R^n$ be some set of directions. A set $X\subseteq\mathbb R^n$ is called $V$-convex if the intersection of any line parallel to a vector in $V$ with $X$ is connected. Semispaces and the problem of the least intersection base for partial convexity is investigated. The cone of convexity directions is described for a closed set in $\mathbb R^n$. 
@article{MZM_1996_60_3_a7,
     author = {N. N. Metel'skii and V. N. Martynchik},
     title = {Partial convexity},
     journal = {Matemati\v{c}eskie zametki},
     pages = {406--413},
     publisher = {mathdoc},
     volume = {60},
     number = {3},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_3_a7/}
}
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N. N. Metel'skii; V. N. Martynchik. Partial convexity. Matematičeskie zametki, Tome 60 (1996) no. 3, pp. 406-413. http://geodesic.mathdoc.fr/item/MZM_1996_60_3_a7/