# Do radially bounded sets form a bornology?

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?

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.