Let $A$ be a non-empty convex subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d \setminus A$, show that there exists $p \in \mathbb{R}^d \setminus \{0\}$ such that $$p \cdot x \leqslant p \cdot y, \forall y \in A.$$
Let $A$ be a non-empty convex subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d \setminus A$, show that there exists $p \in \mathbb{R}^d \setminus \{0\}$ such that
$$p \cdot x \leqslant p \cdot y, \forall y \in A.$$