convexity - If a quadratic form is positive definite on a convex set, is it convex on that set?

Consider a real symmetric matrix $Ainmathbb{R}^{n times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A x geq 0$ for all $x in mathbb{R}^n$.

Now suppose we have a convex subset $Phi$ of $mathbb{R}^n$ such that $x in Phi$ implies $x^T A x geq 0$. Is $x^T A x$ a convex function on $Phi$ (even if $A$ is not positive definite)? Of course, the answer in general is "no," but we can still ask about the most inclusive conditions under which convexity holds for a given $A$ and $Phi$. In particular I'm interested in the question:

Suppose we have a quadratic form $Q:mathbb{R}^{n times n} rightarrow mathbb{R}$. What is the weakest condition on $Q$ that guarantees it will be convex when restricted to the set of positive semidefinite matrices?