Gaussian expectation restricted to a convex polytope

Question

Let $X$ be a Gaussian vector in $\mathbb{R}^n$ with $\mathbb{E}[X]=0$ and $\mathbb{E}[X X^\intercal]=I_n$. Let $\mathbf{S}$ be a convex polytope in $\mathbb{R}^n$ defined as the intersection of $m$ $(m<n)$ half spaces, $\{ x\in\mathbb{R}^n\mid \langle x,\theta_i\rangle\ge 0 \}_{1\le i\le m}$. How do we compute $\mathbb{E}[|X|^2 \mathbb{1}_{X\in \mathbf{S}}]$?

I believe when $m=2$, there is a clean explicit formula for the expectation and it should depend on $n$ and the angle between $\theta_1$ and $\theta_2$. When $n$ is larger, can we still get an explicit formula for this expectation? If we can, how does it depend on $n$ and the angles?

Answer 1

$\newcommand\R{\mathbb R}$Let $S:=\mathbf S$. Let $X$ be any random vector in $\R^n$ with a spherical symmetric distribution (say such that $P(X=0)=0$). Then $|X|$ is independent of $U:=X/|X|$ and $U$ is uniformly distributed on the unit sphere. So, $$E|X|^2\,1(X\in S)=E|X|^2\,1(U\in S)=E|X|^2\,P(U\in S)=E|X|^2\,p_{n,S},$$ where $p_{n,S}:=P(U\in S)$; the displayed equalities actually hold even if $P(X=0)\ne0$. (If $X$ is a standard Gaussian vector in $\R^n$, then of course $E|X|^2=n$.)

As noted by Ruben, p. 213, "For dimensionality greater than three (spherical tetrahedra, spherical pentahedra, etc.) the areas [that are the probabilities $p_{n,S}$ -- I.P.] can no longer be expressed in terms of elementary functions." For such dimensions, there are only recursive formulas expressing $p_{n,S}$ in terms of integrals of $p_{n-2,S'}$ for some polyhedral cones $S'$ in $\R^{n-2}$. These recursive formulas were first obtained by Schläfli in 1858; see e.g. Plackett for a generalization and further references.