Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 208 / 2 / d

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- Solution d

Solution

The sharp Poincaré inequality for the uniform distribution on an interval is

Var (f (U)) \leq \frac{1}{π ^{2}} E [f^{'} (U)^{2}], U \sim Unif [0, 1] .

Tensorizing these

n

identical one-dimensional inequalities gives

Var (f (X)) \leq \frac{1}{π ^{2}} E ∥ \nabla f (X) ∥^{2}

for the uniform distribution on

[0, 1]^{n}

. Therefore one may take

c = 1/ π

; this value is sharp, as functions depending only on one coordinate attain the one-dimensional constant.

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