Example 13.1.22 (Separated).label Consider
\begin{align*}A=\curl{(x,e^x:x\in\R)}&&B=\curl{(x,0):x\in\R}\end{align*}
then $A\cap B=\emp$. $A$ and $B$ are weakly separated since
\begin{align*}\inner{(x_1,x_2)}{\UBC{(0,1)}{a}}&=\UBC{0}{\al}\\ \inner{(x,e^x)}{(0,1)}&=e^{x}>0\\ \inner{(x,0)}{(0,1)}&=0\leq 0\end{align*}
However $A$ and $B$ are not strongly separated since they ”touch” (the sup and inf approaches each other) at $-\infty$. We can change $A=\curl{(x,y):y\geq e^x}$ to get $A,B$ convex and in context of Theorem 13.1.26 $A,B$ are both not bounded.
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