A question on convex hull Let $a_1, a_2,\ldots, a_n$ be $n$ points in the $d$-dimensional Euclidean space. Suppose that $x$ is a point which does not belong to the convex hull of $a_1, a_2,\ldots, a_n$.
My question is, does there exist a vector $v$ such that $\langle v, x-a_i\rangle < 0$ for all $i$? My geometric intuitions tell me that it is true, but cannot find a rigorous proof. A rigorous proof will be appreciated!

Let $y$ be the point on the hull closest to $x$. Then $v=x-y$ works.
The intuition is that all points in the hull are on the same side with respect to the hyperplane passing through $x$ and orthogonal to $x-y$. 
