How to get optimal bound for $\int_{B(d,\mathbf{m})} p(\mathbf{x}) d\mathbf{x}$ for normal distribution $p$ and other bell-shape distributions?

Question

For a multivariate normal distribution $p$, how do we get an optimal bound for $\int_{B(d,\mathbf{m})} p(\mathbf{x}) d\mathbf{x}$?

Please note the the important constraint is $\mathbf{m} \neq 0$ and $d>1$ with $\mathbf{x} \in \mathbb{R}^d$. ($d=1$ case is trivial.)

Note that the mean is set $0$ and $\Sigma = \sigma^2 I$ so that it has symmetry along all directions.

And if there is any standard technique to solve the above, can we generalize the technique to other bell-shape distributions centered at zero?

So as I understand it the Gaussian is centred at the origin, and you are interested in the integral over the ball entered at some arbitrary point $m$? The Cameron-Martin theorem may be of use to you — 2017-02-28
@Nadiels Thanks for the suggestion. After posting it, I've realized the Gaussian case might be easier than general bell-shape distributions because the distribution can be split into $p(\mathbf{x}) = p(x_1) \cdots p(x_d)$. For general bell-shape distributions, hopefully the theorem you suggested work — 2017-02-28

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