Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that $$ V \geq \frac{n^{n/2}(n+1)^{(n+1)/2}}{n!} r^n $$ with equality if and only if $\Delta^n$ is regular (e.g., Mitrinovic, Pecaric, and Volenec (1989) Recent Advances in Geometric Inequalities, pp. 501).

This means that, fixing the inradius, the $n$-simplex that minimizes its (Lebesgue) volume is uniquely given by the regular $n$-simplex.

Now, I have a feeling that the same conclusion holds true for every positive measure that is rotation-invariant about the origin (which is taken to be the center of the inscribed sphere).

How can I prove (or disprove) this claim?

Even if I can only prove it for the standard Gaussian measure, it will be very much appreciated!

Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that $$ V \geq \frac{n^{n/2}(n+1)^{(n+1)/2}}{n!} r^n $$ with equality if and only if $\Delta^n$ is regular (e.g., Mitrinovic, Pecaric, and Volenec (1989) Recent Advances in Geometric Inequalities, pp. 501).

This means that, fixing the inradius, the $n$-simplex that minimizes its (Lebesgue) volume is uniquely given by the regular $n$-simplex.

Now, I have a feeling that the same conclusion holds true for every positive measure that is rotation-invariant about the origin (which is taken to be the center of the inscribed sphere).

How can I prove (or disprove) this claim?

Even if I can only prove it for the standard Gaussian measure, it will be very much appreciated!

Edit: I do not need uniqueness. Trivially, if one considers a measure that assigns zero to every set, then the uniqueness is violated.