Do radially bounded sets form a bornology?

Question

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.

Answer 1

Working in $\mathbb{R}^2$ and using polar coordinates with angle in $[-\pi,2\pi)$, let $$X=\left\{(r,\theta): \left(\theta\not=0\implies r<{1\over \vert\theta\vert}\right) \wedge(\theta=0\implies r=0)\right\}.$$ Then $X$ is radially bounded but the convex hull of $X$ is not radially bounded, and the sum of $X$ with itself is not radially bounded either.