Disjoint convex sets that are not strictly separated Question 2.23 out of Boyd & Vanderberghe's Convex Optimization:

Give an example of two closed convex sets that are disjoint but cannot be strictly separated.

The obvious idea is to take something like unbounded sets which are disjoint but approach each other in the limit. For example, $f(x) = \frac1x$ and $g(x) = -\frac1x$. But isn't $x=0$ a strictly separating hyperplane here?

Take $X = \{(x,y) \mid xy\geq 1, x,y>0\}$ and $Y = \{(x,y) \mid x\leq 0\}$.
