Geodesic
Distribution of the volume of weighted Gaussian simplex
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 187-197

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Let $X_0, \ldots, X_l$ be independent standard Gaussian vectors in $\mathbb{R}^d$ such that $l \leqslant d$. We derive an explicit formula for the distribution of the volume of weighted Gaussian simplex without the origin — $l$-dimensional simplex $\mathrm{conv}(\sigma_0X_0, \ldots, \sigma_lX_l)$ ($\sigma_0, \ldots, \sigma_l > 0$). 
@article{ZNSL_2020_495_a10,
     author = {T. D. Moseeva},
     title = {Distribution of the volume of weighted {Gaussian} simplex},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {187--197},
     publisher = {mathdoc},
     volume = {495},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a10/}
}
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T. D. Moseeva. Distribution of the volume of weighted Gaussian simplex. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 187-197. http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a10/