If a linear objective function over a constriant can attain a finite maximum, then it must be able to obtain this maximum on a boundary point

Question

Let $C\subset \Bbb R^n $ be a non-empty constraint (may be non-convex) and $f({\mathbf x}) = {\mathbf w}^{\operatorname{T}}{\mathbf x}:C \to \Bbb R$ be a linear real-valued objective function. If $f$ obtains a finite maximum at some ${\mathbf x}^*\in C$, is there always a boundary point ${\mathbf x}_b^* \in \partial C$ such that $f$ also obtains maximum at ${\mathbf x}_b^*$ (i.e. ${\mathbf w}^{\operatorname{T}}{\mathbf x^*}={\mathbf w}^{\operatorname{T}}{\mathbf x_b^*}=\max_{{\mathbf x}\in C} {\mathbf w}^{\operatorname{T}}{\mathbf x}$)?

The definition of a boundary point can be found here http://mathworld.wolfram.com/BoundaryPoint.html.

Answer 1

Restrict to the one-dimensional subspace $\mathbf x^*\Bbb R$. In the one-dimensional case the claim is clear.