Quadratic form constant over constrained region

Question

For all vectors $x$ such that the $i$th element of $x$, $x_i \in [-1, 1]$ $$x^TAx = c > 0$$ for known positive constant c. Assuming we know nothing about $A$ except that it is symmetric, when is this possible? I know $A$ must be either positive definite, positive semi-definite, or indefinite (since we know nothing about $A$ when elements of $x$ are outside the constrained region). Can we say anything else about $A$ in this situation?

Answer 1

Let $x \in \mathbb{R^n}$ such that $x_i \notin [-1,1]$ for some $i$. If $y=\frac{x}{\| x\|}$ then $y_i \in [-1, 1]$ for all $i$, hence $$y^TAy=c \iff \frac{1}{\| x\|^2}x^TAx=c \iff x^TAx=c\| x\|^2 > 0 $$

and $A$ is positive definite.