Can someone clarify the following definition for me.
Let $V$ be a real-valued vector space. A convex set $S$ in $V$ is called linearly bounded when every straight line intersects $S$ in a bounded subset of that line.
Can someone please give me an example of such a set and a counter example. I have google around but can not find an example.
In $\ell^2$, define $C$ to be the convex hull of all sequences $(x_n)$ such that $(\forall n)$ $|x_n|\leq n$. This is clearly an unbounded set. However, it is linearly bounded. See https://www.univie.ac.at/EMIS/journals/HOA/AAA/Volume2003_2/620817.pdf and the references therein.