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Let $K\subset\mathbb{R}^d$ be a nonempty convex subset s.t $0\not\in K$. Then there exists $\phi\in\mathbb{R}^d$ with $\phi\cdot k\geq 0$ for all $k\in K$ and $\phi\cdot k_0>0$ for some $k_0\in K$. Is it in general possible to have such a $\phi$ in the half space $H=\{x\in\mathbb{R}^d : \sum_{i=1}^d x_i \leq 0\}$?

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No. Already in dimension $d = 1$, where $H = (-\infty,0]$: Consider any non-empty convex subset $K \subset (0,\infty)$.