I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$How can I compute probability densities within the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure? ${}$
The closest question & answer I have found on this site was for a particular instance of Gaussian and hyperplane here.
$\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}$First, one should not denote a random vector in $\R^n$ (which is not actually a vector in $\R^n$ but a function with values in $\R^n$) and a true, non-random vector in $\R^n$ by the same symbol, such as $\mathbf x$.
Accordingly, let $X\sim N(0,S)$, where $S:=H^{-1}$. Let $c:=C$ and $g:=\mathbf g\ne0$. We want to find the conditional density of $X$ given $X^\top g=c$; as usual, we identify $\R^n$ with the set $\R^{n\times1}$ of all $n\times1$ real matrices. Without loss of generality, $g$ is a unit vector; otherwise, replace $g$ and $c$ by $g/|g|$ and $c/|g|$, respectively, where $|g|$ is the Euclidean norm of $g$. Let $g_1,\dots,g_n$ be any orthonormal vectors in $\R^n$ such as $g_n=g$.
Let $Y_1,\dots,Y_n$ be the coordinates of the random vector $X$ in the orthonormal basis $(g_1,\dots,g_n)$, so that $Y_k=g_k^\top X$ and $X=\sum_{k=1}^n Y_kg_k$. Let $Y:=[Y_1,\dots,Y_n]^\top$, so that $Y=G^\top X$ and $X=GY$, where $G:=[g_1,\dots,g_n]$, the $n\times n$ matrix with columns $g_1,\dots,g_n$. The covariance matrix of $Y$ is \begin{equation} EYY^\top=G^\top SG=: \begin{bmatrix} \Si_{11}&\Si_{12} \\ \Si_{21}&\Si_{22} \end{bmatrix}, \end{equation} where $\Si_{12}$ is the $(n-1)\times1$ matrix that is the covariance matrix of $[Y_1,\dots,Y_{n-1}]^\top$ and $Y_n$. Also, $Y\sim N(0,G^\top SG)$. So, the conditional pdf of $[Y_1,\dots,Y_{n-1}]^\top$ given $Y_n=c$ is the pdf, say $p_c$, of the $(n-1)$-dimensional normal distribution \begin{equation} N(\Si_{12}\Si_{22}^{-1}c,\Si_{11}-\Si_{12}\Si_{22}^{-1}\Si_{21}). \end{equation}
This conditional pdf, $p_c$, can be regarded as the desired conditional density of $X$ given $X^\top g=c$. Indeed, $p_c$ can be used as follows: for any (say) nonnegative Borel function $f\colon\R^n\to\R$, \begin{equation} E(f(X)|X^\top g=c)=E(f(GY)|Y_n=c) \\ =\int_{\R^{n-1}}f(G[y_1,\dots,y_{n-1},c]^\top)p_c(y_1,\dots,y_{n-1})\, dy_1\cdots dy_{n-1}. \end{equation} In particular, for any Borel subset $B$ of $\R^n$, \begin{equation} P(X\in B|X^\top g=c)=P(GY\in B|Y_n=c) \\ =\int_{\R^{n-1}}1(G[y_1,\dots,y_{n-1},c]^\top\in B)p_c(y_1,\dots,y_{n-1})\, dy_1\cdots dy_{n-1}. \end{equation}
