Given a real vector space $X$, we can look at subsets $S \subset X$ where, for each $x \in S$, there exists a $v \in X$ such that $x+av \in S$ for all non-negative $a$. That is to say, $S$ has the property that it contains at least one ray based at each of its points. A less-precise way of thinking about this is that $S$ is "star-convex at the point at infinity".
The motivation for such sets is a model for a constrained sort of constructive solid geometry, possibly modeling certain kinds of CNC-constructability.
Is there a name for such sets? Some obvious basic observations:
- The set of such subsets of a space $X$ is closed under unions.
- The complement of a star-convex set clearly has this property. (Proof: take $v=x-x_0$ where $x_0$ is a center for the star-convex set.)
- There are sets with this property whose complements are not star-convex. (Example: a U-shaped set.)
Are there other interesting properties, like relationships or equivalence to other classes of sets?
These sets can be characterized as the (possibly uncountable) unions of rays. On the one hand, any set with this property can be written as $$ S =\bigcup_{x \in S} \{x + a v_x : a\ge 0\} $$ where $\{x + a v_x : a\ge 0\}$ is the ray starting at $x$ contained in $S$ that we know has to exist. On the other hand, all rays (and therefore all unions of rays) have the property you want.
Without assuming any other properties, there isn't anything more we can say about such sets.