Minimizing a linear function over an affine set

Question

Consider the following. Let $A\subseteq \mathbb{R}^n$ be an affine set, $c\in \mathbb{R}^n$. Consider the optimization problem $\min_{x\in A} c^T x$. And suppose there exists a $x^*\in A$ that achieves the minimum. Is it true that all $x\in A$ are also achieve the minimum, i.e., $c^Tx^*=c^Tx$ for all $x\in A$? I read a proof that seemed to suggest this result, but I have tried and could not prove it. Suggestions?

Answer 1

Think about the 1-dimensional function $x \mapsto cx$. It only has a minimum if $c$ is $0$, that is, when it's a constant function on the domain. This generalizes to higher dimensions.

If a linear function is not constant over an affine set, then it is unbounded. Suppose $x,y\in A$ and $c^Tx < c^Ty$, then $x + t(x-y) \in A$ and for all $t$ and $\lim_{t\rightarrow \infty} c^T(x + t(x-y)) = -\infty$. So if $x^*$ achieves a minimum, then the function must be constant over $A$, meaning $c^T x^* = c^Tx$ for all $x \in A$.