Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 4 / b / Solution

We prove the Convex Poincaré inequality. For a differentiable convex function $h:[0,1]\to \mathbb{R}$ and independent copies $U,U^{\prime }$ supported on $[0,1]$, convexity givesTaking expectations and using $\mathrm{Var}(h(U))=\frac{1}{2}\mathbb{E}(h(U)-h(U^{\prime }){)}^2$ givesApplying this conditional inequality coordinate by coordinate in the Efron–Stein inequality provesSince $-g$ is convex and has the same gradient norm as $g$, the same argument gives $\mathrm{Var}(g(X))\le 1$.