Suppose $g:\mathbb{R}^n\to\mathbb{R}$ is twice differentiable continuously function in $\mathbb{R}^n$ and suppose that $g$ is strictly convex in non-convex OPEN set $D\subset \mathbb{R}^n$. I would like to ask whether local minimum of $g$ in $D$ must be global in $D$.
Namely, suppose $u\in D$ is local minimum of $g$, does $ g(u)\le g(x) $ for all $x\in D$ ?
No. A simple counter example: $$f(x)=x^2 \\ D = \{-3\leq x\leq -2, 1\leq x\leq 2\}$$ Then $x=-2$ is a local minimum, but $x=1$ is the global minimum.