[vrrofacp] Weaker than convexity

A geometrically convex set has that a line segment between any two points within the set is also entirely in the set.

Weaker condition: there exists a point, a "center", such that a line segment from the center to any point in the set is contained in the set.  Does this property have a name?  Vaguely similar to the original spline.

Inspiration is that, for a polyhedron with this property, one can compute its volume by summing pyramids with the apex at the center without having to think hard about double counting due to faces which are facing backwards.

Even weaker: union of several sets each with different centers.  The set of centers could be an interesting geometric object, e.g., a line.

Update: answer is "star convexity". Star Domain at Wikipedia. (Thanks anonymous commenter.)