# A property of convex cones in Euclidean spaces

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.

• Don't you mean "$L\subseteq B^n$"? – Matemáticos Chibchas Jul 5 '19 at 15:01
• Could $K$ be all of Euclidean space? – Ben McKay Jul 5 '19 at 15:03
• @BenMcKay Thanks, corrected. – MKO Jul 5 '19 at 15:36