Consider a smooth function $F : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ with $m,n > 1$.
Definition: $(x^* , y^*) \in \mathbb{R}^n \times \mathbb{R}^m$ is a saddle point of $F$ if there exist open sets $O_1 \subset \mathbb{R}^n$ and $O_2 \subset \mathbb{R}^m$ such that $ F(x^* , y) \leq F(x^* , y^*) \leq F(x,y^* ) \quad \forall \: x \in O_1, \: y \in O_2 $.
Denote the Hessian of $F(\cdot,y^*)$ evaluated at $x^*$ by $H_1$ and the Hessian of $F(x^*,y)$ evaluated at $y^*$ by $H_2$. If the gradient of $F$ at $(x^*,y^*)$ is zero and $H_1$ is positive definite and $H_2$ is negative definite, does it imply that $(x^* , y^*)$ is a saddle point?
My attempt at establishing the implication is as follows. Note that the gradient of $F$ at $(x^*,y^*)$ is zero. Now, if H1 is positive definite, $F(\cdot,y^*)$ has a strict local minimum at $x^*$ thus establishing the first part of the inequality. Similarly, if H2 is negative definite, $F(x^*,\cdot)$ has a strict local maximum at $y^*$ thus establishing the second part of the inequality.