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Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$. It is not hard ...