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Expectation of probability on simplex at least 1/(d+2)

Source: Miklós Schweitzer 2016, Problem 9

November 2, 2016
probability and statsprobability distributionprobabilitycollege contestsMiklos Schweitzer

Problem Statement

For p0,…,pd∈Rdp_0,\dots,p_d\in\mathbb{R}^d, let S(p0,…,pd)={α0p0+⋯+αdpd:αi≤1,∑i=0dαi=1}. 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 π\pi be an arbitrary probability distribution on Rd\mathbb{R}^d, and choose p0,…,pdp_0,\dots,p_d independently with distribution π\pi. Prove that the expectation of π(S(p0,…,pd))\pi(S(p_0,\dots,p_d)) is at least 1/(d+2)1/(d+2).
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