I would like some input on if my understanding and solution to the following problem is correct:
Given a spherical normal distribution $N(\mu, \Sigma)$ of dimension $d$ with:
$$\Sigma = \left[ \begin{array}{cccc} \sigma_1 ^2 & 0 & \dots & 0\\ 0 & \sigma_2 ^2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \sigma_d ^2 \end{array} \right] \qquad \sigma = \sigma_1 = \dots = \sigma_d $$
We select $n$ variables $K_n = \{X_1, \dots, X_n\}$ with each $X_i \sim N(\mu, \Sigma)$ independent and identically distributed from this distribution, and wish to calculate the expected volume of the convex hull $E[Vol(K_n)]$.
From F. Affentranger's work (http://www.seminariomatematico.unito.it/rendiconti/cartaceo/49-3/359.pdf) in 1991 we have that if $N(\mu, \Sigma) = N(0, 1)$ is the standard spherical normal distribution of unit variance in dimension $d$, then the expected volume of the convex hull of $n$ random points selected from this distribution is:
$$E[Vol(K_n)] = \kappa_d (\log n)^{d / 2} (1 + o(1))$$
where $\kappa_d$ is the volume of the $d$-dimensional unit ball.
Given that arbitrary normal distributions $N(\mu, \sigma ^2)$ can be reduced to the standard normal distribution $N(0, 1)$ by the relation:
$$ N(\mu, \sigma ^2) = \sigma N(0, 1) + \mu $$
Is my understanding correct when I claim that it follows that the convex hull of $n$ randomly selected points $K_n$ from an arbitrary spherical normal distribution $N(\mu, \Sigma)$ with $\Sigma_{i, i} = \sigma^2, \Sigma_{i, j, i \neq j} = 0$ would be:
$$ \begin{align*} E[Vol(K_n)] &= E\big[Vol(\{X_1, \dots, X_n\})\big] \\ &= E\big[Vol(\{\sigma N(0, 1)_1 + \mu, \dots, \sigma N(0, 1)_n + \mu\}) \big] \\ &= E\big[ Vol( \{ \sigma N(0, 1)_1, \dots, \sigma N(0, 1)_n \}) \big] \\ &= \sigma^d E\big[ Vol( \{ N(0, 1)_1, \dots, N(0, 1)_n \}) \big] \\ &= \sigma^d \big[\kappa_d (\log n)^{d / 2} (1 + o(1)) \big] \end{align*} $$