For standard Gaussian measure $\gamma$, the <Gaussian Poincaré inequality> is
$$
\operatorname{Var}_\gamma h\leq\int(h')^2d\gamma.
$$
The <Gaussian logarithmic Sobolev inequality> is
$$
\operatorname{Ent}_\gamma(h^2)
\leq2\int(h')^2d\gamma,
$$
where $\operatorname{Ent}_\gamma(G)=\int G\log G\,d\gamma-(\int Gd\gamma)\log\int Gd\gamma$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 3 / a / Solution

With the same $h$, the squared <Hellinger distance> is
$$
d_H(f,\phi)^2=\frac12\int(h-1)^2d\gamma
=1-\int h\,d\gamma.
$$
Since $0\leq\int h\,d\gamma\leq1$,
$$
1-\int h\,d\gamma
\leq1-\left(\int h\,d\gamma\right)^2
=\operatorname{Var}_\gamma h.
$$
The <Gaussian Poincaré inequality> and the derivative calculation in part (b) yield
$$
d_H(f,\phi)^2\leq\frac14J(f\Vert\phi).
$$

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 3 / c / Solution

{c}
{wiki}

For standard Gaussian measure $\gamma$, the sharp Poincaré constant is one:
$$
\operatorname{Var}_\gamma f\leq\int\lVert\nabla f\rVert^2d\gamma.
$$


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