EDIT: Let $K$ be a closed convex cone in a Euclidean space of finite dimension. Assume it is non-zero and not the whole space.

Does there exist a non-zero point $x\in K$ such that $$(x,y)\geq 0 \mbox{ for all } y\in K?$$

An equivalent reformulation of the problem: Given a convex compact subset $L\subset S^{n-1}$ of the unit Euclidean sphere. Does there exist a spherical ball of radius at most $\pi/2$ containing L?

Remark. In dimension 2 the answer is positive.