Kahane-Khintchine inequality

For a Rademacher sum in a normed vector space, any two fixed positive moments of its norm are comparable with constants depending only on those moments. The sharp Rademacher second-moment inequality gives the sharp first-to-second moment comparison $\Vert S{\Vert }_{L^2}\le \sqrt{2}\Vert S{\Vert }_{L^1}$ for arbitrary coefficients.

For $S={\sum }_i{a}_i{ϵ}_i$ and $F=\Vert S\Vert$, the even-function spectral gap on a hypercube gives $2\mathrm{Var}(F)\le -\mathbb{E}[FLF]$. A norming functional supplied by the Hahn-Banach theorem and convexity of the norm gives $LF\ge -F$, because $LS=-S$. Hence the Dirichlet form is at most $\mathbb{E}{F}^2$, proving the inequality. Two equal scalar coefficients attain equality, so the constant is sharp.