I need your expertise in proving the following:
Given a convex body $C \subseteq \mathbb{R}^n$, and a unit vector $a \in \mathbb{R}^n$, we wish to prove that the point that maximizes $f(x) = c^Tx$ is a boundary point, i.e.,
$$\begin{align} x_0 := \arg\max_{x \in C} a^Tx \in \partial(C) \end{align}$$
How can we prove it? Intuitively speaking, it's easy to prove by contradiction, but we wonder whether there is an actual theorem or lemma out there (there should be) that allows one to prove the above.
Please advise and thanks in advance.
The Linear function $f$ attains its max/min at an interior point on a constraint set if and only if $f=0$ . (Proof is elementary )
Therefore, if $f \neq 0$ is a Linear function, and $C$ is a (convex) compact set in $R^n$ then $f$ attains its maximum on the boundary of $C.$
P.s Convexity is not needed here ! but compactness is essential .