Show that $S := \{p \in \Delta^n \mid b^T p + p^T Ap \leq \alpha\}$ is a convex set where $\alpha \in \mathbb{R}$, $A\succeq 0$, $\Delta^n$ is the probability simplex.
I have been stuck for this for hours. My main approach is to take two points $p, q \in S$ and show that $\lambda p + (1-\lambda) q \in S$, but the formula becomes very complicated to manage.
I am wondering if there is an easier approach.