Partial convexity

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 $\mathbb R^n$.