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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.

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  • $\begingroup$ Don't you mean "$L\subseteq B^n$"? $\endgroup$ – Matemáticos Chibchas Jul 5 '19 at 15:01
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    $\begingroup$ Could $K$ be all of Euclidean space? $\endgroup$ – Ben McKay Jul 5 '19 at 15:03
  • $\begingroup$ @BenMcKay Thanks, corrected. $\endgroup$ – MKO Jul 5 '19 at 15:36
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Follows from the hyperplane separation theorem (https://en.wikipedia.org/wiki/Hyperplane_separation_theorem).

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