= Strict increase of a Gaussian norm distribution
Suppose $B$ is a nonzero <separable Banach space>, the <Cameron-Martin space of a Gaussian random variable in a Banach space> is a <dense subset> of $B$, and every centered ball of positive radius has positive <probability>. Then $t\mapsto\mathbb P(\|X\|_B\leq t)$ is strictly increasing on $(0,\infty)$. For $0<a<b$, choose $h$ in that <Cameron-Martin space of a Gaussian random variable in a Banach space> with $\|h\|_B=(a+b)/2$ and $0<\delta<(b-a)/2$. The closed ball of radius $\delta$ centered at $h$ lies in the annulus $a<\|x\|_B<b$ and has positive <probability> by the <symmetric Gaussian translation lower bound>. The nonzero assumption is necessary: on $B=\{0\}$ the distribution function equals one everywhere.
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