= Symmetric Gaussian translation lower bound
{title2=$\mathbb P(X-h\in C)\geq e^{-\|h\|_H^2/2}\mathbb P(X\in C)$}
For a centered <Gaussian random variable in a Banach space>, a <Borel set> $C=-C$, and $h$ in its <Cameron-Martin space of a Gaussian random variable in a Banach space>, the <Cameron-Martin theorem for a Gaussian measure> gives the translated density $e^{-\widehat h(X)-\|h\|_H^2/2}$. The joint law of $(X,\widehat h(X))$ is invariant under simultaneous negation. Averaging the two density formulas therefore replaces $e^{-\widehat h(X)}$ by $\cosh\widehat h(X)\geq1$, proving the lower bound. No <convexity> of $C$ is required.
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