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TIETZE-TYPE THEOREM FOR PARTIALLY CONVEX PLANAR SETS
Let S be a nonempty subset of R2 and R2 a set of directions. S is called V-convex or partially convex relative to at a point s clS if and only if there exists a neighbourhood N of s in R2 such that the intersection of any straight line parallel to a vector in with S N is connected or empty S is called -convex or partially convex relative to if and only if the intersection of any straight line parallel to a vector in with S is connected or empty. It is proved that if is open, S is connected and open or polygonally connected and closed, and -convex at every boundary point, then it is -convex. This contributes to a recent work of Rawlins, Wood, Metelskij and others.
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