I have read the following claim
The difference of two closed convex cones in $\mathbb R^n$ can be non closed
but I am not convinced and cannot manage to find a counter-example, can you find any?
By difference of sets $A,B$ I mean the algebraic difference: $$A-B:=\{x\in \mathbb R^n: \exists a\in A,b\in B,\ \ x=a-b\}$$
Choose in $\mathbb{R}^3$ two circular cones $C_1$ and $C_2$ with common zero apex, which touch each other along a halfline, say $h$. Denote by $H$ the plane separating $C_1$ and $C_2$. Then $C_1 - C_2$ is the convex set consisting of the open halfspace (bounded by $H$ and containing $C_1$ and $-C_2$) and the line $h \cup (-h)$.