= Sharp Rademacher second-moment inequality
{title2=$\mathbb E\|S\|^2\leq2(\mathbb E\|S\|)^2$}
For $S=\sum_i a_i\epsilon_i$ and $F=\|S\|$, the <even-function spectral gap on a hypercube> gives $2\operatorname{Var}(F)\leq-\mathbb E[FLF]$. A norming functional supplied by the <Hahn-Banach theorem> and convexity of the <norm> gives $LF\geq-F$, because $LS=-S$. Hence the Dirichlet form is at most $\mathbb EF^2$, proving the inequality. Two equal scalar coefficients attain equality, so the constant is sharp.
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