Is $q(x) = \frac{1}{2} x^T H x - g^T x$ strongly convex?
Let $$q(x) := \frac{1}{2} x^T H x - g^T x$$ where matrix $H$ is symmetric positive definite and $g \in \mathbb{R}^n$. Clearly, the function is strictly convex, but why is it strongly convex?
Yes, $q$ is $\alpha$-strongly convex where $\alpha$ is the minimum eigenvalue of $H$. We have that $\alpha > 0$ because $H$ is positive definite. Notice that $q$ is $C^2$ and $\nabla^2 q = H$. Since a $C^2$ function is $\alpha$-strongly convex iff its Hessian has all its eigenvalues above $\alpha$, we are done.