If the determinant of Hessian is negative, what can we say about the matrix?

Question

This question is based on this answer for the question How do I prove that this objective function is not convex?

---

The objective expression becomes $2y^2x^2$. The Hessian of this expression is $$\begin{bmatrix} 4y^2 & 8xy \\ 8xy & 4x^2 \end{bmatrix}$$ The determinant of this Hessian is $-48x^2y^2$, which is negative when both $x$ and $y$ are nonzero, so it cannot possibly be positive semidefinite. Hence the objective function is neither convex nor concave.

---

I don't understand the last line of this answer where author says "Hence the objective function can is neither convex nor concave." In other words, can someone fill in these blanks ?

If determinant of hessian matrix of $\mathbf{A}$ is

1. POSITIVE, $\mathbf{A}\in S^{N}_{++}$ and objective function is CONVEX

2. ZERO, $\mathbf{A} \in$ ___ and objective function is _________

3. NEGATIVE, $\mathbf{A} \in$ ___ and objective function is _________

Answer 1

Hint for 3: What is the opposite of convex?

Hint for 2: If a point is neither convex nor concave, it must be convex in one direction and concave in another (or flat).