SCHWEITZER 2016 Problem 9

For $p_0,\dots,p_d\in\mathbb{R}^d$ , let $$S(p_0,\dots,p_d)=\left\{ \alpha_0p_0+\dots+\alpha_dp_d : \alpha_i\le 1, \sum_{i=0}^d \alpha_i =1 \right\}.$$ Let $\pi$ be an arbitrary probability distribution on $\mathbb{R}^d$ , and choose $p_0,\dots,p_d$ independently with distribution $\pi$ . Prove that the expectation of $\pi(S(p_0,\dots,p_d))$ is at least $1/(d+2)$ .