Exercise 2.15g of Boyd & Vandenberghe's Convex Optimization:
On the probability simplex in $\mathbb{R}^n$ where each point $p = (p_1, p_2, p_3, \ldots, p_n)$ corresponds to a distribution for random variable $X$ to take one of $n$ values: $\mathbb{P}(X = a_i) = p_i$ where $a_j < a_k$ for $j < k$. Is the region where $\mbox{Var}(X)\geq\alpha$ for some positive $\alpha$ a convex region or not?
This has a solution on the Stanford SEE course material of Prof Boyd's course but I didn't follow the reasoning. Would appreciate help on the answer.