A version of Hahn-Banach separation theorem

Question

Let $X\subset\mathbb R^n$ be a closed convex set and let $L$ be a line not intersecting $X$. Is there a hyperplane containing L and not intersecting $X$?

Thanks in advance.

Answer 1

Yes. There is a separation theorem stating that two convex sets in $\mathbb{R}^n$ such that the relative interior of one of them does not intersect the other can be separated by a hyperplane (this is in J. van Tiel's book on Convex Analysis, for example). The theorem applies to your set and your line. Such hyperplane has to be parallel to the line of course. Now just shift your hyperplane parallel to itself until it contains the line. The separation is not necessarily strict unless your convex set is compact.