Let's say that A is a convex set in $R^2$. Now assume that L is a line in $R^2$.
$L=\{x: p\cdot x = t\}$ where p and x are both contained in $R^2$, $p\cdot x$ is the inner product of p and x, and t is any number in R.
Also A and L do NOT intersect. It seems obvious to me that A is within an open halfspace of L, but I'm not sure what a proof might look like. I guess I'd have to show that all points in A will be greater than or less than an equivalent point in L, but I haven't a clue how to go about it. Any help? Thanks.