Let $C \subset \mathbb{R}^n$ be convex and closed, $y \in \partial C$.
Show that there is an affine $f(x)=\langle x-y,e \rangle$ where $|e|=1$, such that $f(y)=0$ and $\forall x \in C: f(x) \leq 0$
Notation: Let me explain my notation, in case there is some confusion: $\langle -,- \rangle$ is the euclidean scalar product and $|\cdotp |$ is the derived norm.
Also I want to add some things I already know that could be interesting here: (1) $\forall v \in \mathbb{R}^n \backslash C: ! \exists c \in C \text{such that} |v-c|=\min_{x \in C}|v-x|$
(2) For $v$ and $c$ like in (1): $\forall x \in C: \langle x-c,\frac{v-c}{|v-c|} \rangle \leq 0$
Geometrically if I am not completely wrong you could imagine $e$ as standing orthogonally on $y$ so the scalar product would be always negative. (Sorry for the awful formatting, I will try to improve my latex-skills! If something is not clear, don't hesitate to ask.)