I'm trying to understand what is the correct way to define the PDF of a multivariate truncated Gaussian that is supported on a closed ball, as I find the literature in this subject very limited.
Suppose that $\mathbf{d}\in\mathbb{R}^n$ is a multivariate $n$-dimensional truncated Gaussian with $n\times n$ covariance matrix $\boldsymbol{\Sigma}$ and $\mathbf{a}\in\mathbb{R}^n$ mean. Also assume that this truncated Gaussian is supported on the ball $\mathcal{B}$ with radius $r>0$ centered at $\mathbf{a}\in\mathbb{R}^n$. Let's denote it as $\mathbf{d}\sim\mathcal{N}_{\mathcal{B}}\left(\mathbf{a},\boldsymbol{\Sigma}\right)$.
What is the correct way to write the PDF of this truncated Gaussian? To my understanding it will take the form $$f\left(\mathbf{x}\right)=\frac{1}{Z}\exp\left(-\frac{1}{2}\left(\mathbf{x}-\mathbf{a}\right)^T\boldsymbol{\Sigma}^{-1}\left(\mathbf{x}-\mathbf{a}\right)\right)\cdot I_{\mathcal{B}}\left(\mathbf{x}\right),$$ where $Z>0$ is some positive constant depending on the covariance and the support set ensuring that the integral of $f$ over $\mathcal{B}$ equals $1$, and where $I_{\mathcal{B}}$ is the indicator function over the ball $\mathcal{B}$, i.e., $I_{\mathcal{B}}\left(\mathbf{x}\right)=1$ if $\mathbf{x}\in\mathcal{B}$ and $0$ otherwise.
Is this definition seems correct? In particular, is the indicator function defined correctly? Or should I consider a different indicator that also depends on the covariance matrix? Would you define it differenly?
I'm asking since I only found in the literature definitions of PDFs of truncated Gaussians supported on linearly constrained sets, and not over non-linear sets as a ball. So any help will be much appreciated.