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 $x$?


Here, a vector $v$ is normal to a convex set $C$ at $x$ iff for all $y\in C$, $\langle v,y \rangle\le \langle v,x\rangle$.



If a compact convex set meets the positive orthant does it meet it at a point with a normal in the positive orthant?
While fedja gave an answer in the comments, here is a different approach that yields a more general answer (the non-negative orthant is self-dual).


**Proposition:** Let $C\subseteq \mathbb R^n$ be a compact convex set that meets some closed convex cone $K$. Then $C\cap K$ contains a point with a non-zero normal in the dual of $K$. If $C$ meets the interior of $K$ and the dual of $K$ has non-empty interior, then the point can be taken to have a non-zero normal in the interior of the dual of $K$.


**Proof:** Suppose $K=\{ z : \left< x,z \right> \ge 0 \text{ for all }x\in A \}$ for a countable set $A$ ([this works for any separable Banach space](https://mathoverflow.net/questions/281182/when-are-the-closed-convex-subsets-countable-intersections-of-halfspaces)). Let $B$ be the basis of the largest linear space contained in $K$. Thus if $\left< x,z \right> = 0$ for all $x\in A\cup B$, then $z=0$.


Enumerate $A\cup B=\{a\_1,a\_2,\dots\}$. Define the sequence of non-empty compact sets $C\_i$ by letting $C\_0 = C\cap K$ and $C\_i$ be the set of points of $C\_{i-1}$ that maximize $\left< a\_i, \cdot \right>$. Fix $z\in\bigcup\_i C\_i$.


I claim that $C\cap (z+K) = \{ z \}$. For suppose $w \in K$ and $z+w \in C\cap (z+K)$. The maximality conditions imply that $\left< a\_i, w \right>\le 0$ for all $i$. Hence $w=0$.


Let $H$ be a separating closed half-space with $z$ on its boundary that contains $C$ and the closure of whose complement contains $z+K$. The outward normal of $H$ is then in the dual of $K$.


Now suppose $C$ meets the interior of $K$ at some point $z\_0$ and $K^\*$ has non-empty interior. Let $K\_\epsilon$ be an $\epsilon$-widened closed convex cone, e.g., the cone generated by the compact set $(K\cap H)+\bar B(0;\epsilon)$ (Minkowski sum) where $H$ is a hyperplane through a point in the interior of $K^\*$ and normal to the vector from the origin to that point. Then for sufficiently small $\epsilon>0$, $C\cap (z\_0+K\_\epsilon) \subseteq C\cap \operatorname{Int} K$. Applying the already proved part to $-z\_0+C$ and $K\_\epsilon$, we get a point $z$ in $C$ with a non-zero normal in $(K\_\epsilon)^\*$. But $\operatorname{Int} (K^\*) \supseteq (K\_\epsilon)^\* \backslash \{ 0 \}$.


