Even-function spectral gap on a hypercube
= Even-function spectral gap on a hypercube
{title2=$2\operatorname{Var}(f)\leq-\mathbb E[fLf]\quad(f(-\omega)=f(\omega))$}
Global sign-even functions have zero coefficients on odd-size <Walsh characters>. Every nonconstant remaining character has generator <eigenvalue> at most $-2$, yielding the displayed improvement from gap one to gap two. Combining this with the convexity of a <norm> proves the <sharp Rademacher second-moment inequality>.