= Separation from an absorbing convex set
If a <convex set> $E$ in a real <normed vector space> contains $B(0,\epsilon)$, every $x\notin E$ admits a <continuous linear functional> $T$ with $Tx\geq1\geq Te$ for all $e\in E$, even if $E$ is not closed. Its <Minkowski functional> $p(v)=\inf\{t>0:v\in tE\}$ is <sublinear functional>, satisfies $p(v)\leq\|v\|/\epsilon$, $p(e)\leq1$, and $p(x)\geq1$. Extend $tx\mapsto tp(x)$ by the dominated <Hahn-Banach theorem>. The resulting domination $T\leq p$ applied to $v$ and $-v$ makes $T$ continuous and supplies the separation inequalities.
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