Let $K\subset \mathbb R^n$ be a convex set which is also compact with $dim K=n$ and $H \subset \mathbb R^n$ a hyperplane.
Proof that for any Hyperplane H there exist exactly two supporting hyperplanes of K parallel to H
Can anyone give me a hint how to proof the statement above ?
I know that any given hyperplane can be written as $H=\{x\in R^n : c^tx=\delta \}$ and I know the hyperplane seperation theorem.
Outline:
Assume $K$ is a compact, convex subset of $\mathbb{R}^n$ with $\dim(K)=n$.
Define $f \colon \mathbb{R}^n \to \mathbb{R}$ by $f(x) = c^tx$.
Let $J=f(K)$.
Clearly $f$ is continuous, hence, since $K$ is compact, so is $J$.
Since $K$ is convex and $f$ is linear, $J$ is convex.
Thus, $J$ is a compact, convex subset of $\mathbb{R}$.
Since $\dim(K)=n$, argue that $J$ contains more than one point.
Thus, $J = [a,b]$, for some $a,b \in \mathbb{R}$, with $a < b$.
Argue that $f^{-1}(\delta)$ is a supporting hyperplane if and only if $\,\delta=a\;\,\text{or}\;\,\delta=b$.