# Gaussian integrals over an intersection of two half-spaces

Let $$W_1, W_2 \in \mathbb{R}^{n \times n}$$, $$x, b_1, b_2 \in \mathbb{R}^{n}$$ and $$\sigma \in \mathbb{R}^{+}$$, I am trying to integrate a Gaussian over the intersection of two half-space: $$f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \int_{\Omega_1 \cap \Omega_2} \exp\left(-\frac{\left\Vert x - t \right\Vert_2^2}{2 \sigma^2}\right) d t$$ where $$\Omega_1 = \left\{ z \in \mathbb{R}^{n}: W_1 z + b_1 \geq 0 \right\}$$ and $$\Omega_2 = \left\{ z \in \mathbb{R}^{n}: W_2 z + b_2 \geq 0 \right\}$$

Related questions have been answered here and here but I still haven't figure out a proper way to compute this integral. I also wondering if the result from this paper can be of help.

• I don't think that you have any hope of an analytic expression in terms of elementary functions. There might be something "nice" in terms of the $\operatorname{erf}$ function, but the best that I would expect would be a numerical result. Commented Apr 25, 2023 at 19:34
• One point to simplify matters is to start with lower-dimensional examples. The case of $n=1$ is probably too simple to be interesting, but $n=2$ already looks quite difficult with the range of possible parameters. Commented Apr 25, 2023 at 19:39
• I am not sure why that would be incorrect. In 1 dimension, the set ${z \in \mathbf{R} : w x + b \geq 0 }$ defines a half-space right? Do you mean that $W_1$ and $W_2$ would need to be vectors?