We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, i.e. a collection that contains singletons, and is closed with respect to taking unions and subsets. Moreover, a balanced hull $[-1,1]A$ of a radially bounded set $A$ is radially bounded.
I am wondering whether radially bounded sets form a linear bornology, or generate one in some natural way. This leads to the following questions:
Is the sum of two radially bounded sets radially bounded? What if these sets are convex?
Is convex hull of a radially bounded set radially bounded?
It seems unlikely that this class of sets has not been studied before, so references would be appreciated.