Let $C$ be a compact convex set in $\mathbb R^n$ that intersects the strictly positive orthant $\mathbb R_+^n$. Does $C\cap \mathbb R_+^n$ have to contain a point $x$ such that some vector $v\in\mathbb R_+^n$ is normal to $C$ at $x$?
If a compact convex set meets the positive orthant does it met it at a point with a normal in the positive orthant?
Alexander Pruss
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