Consider the distribution in $\mathcal{D}^{\prime}\left(\mathbb{R}^n\right)$ given, for $R>0$, by
$$
T_R(\phi):=\left|\mathbb{S}^{n-1}\right|^{-1} \int_{\mathbb{S}^{n-1}} \phi(R \omega) \mathrm{d} \omega .
$$
By Theorem 6.22 there exists a unique, regular Borel measure $\mu$ such that $T_R(\phi)=\int \phi(x) \mu(\mathrm{d} x)$.
a) Compute $9.7(1)$ for this measure $\mu$ and compute $D(\mu, \mu)$. You have to show that $|x-y|^{2-n}$ is measurable with respect to $\mu(\mathrm{d} x) \times \mu(\mathrm{d} y)$.
b) Prove that with $\nu(\mathrm{d} x)=\mu(\mathrm{d} x)-\rho \mathrm{d} x$, and $\rho \in L^1\left(\mathbb{R}^n\right)$ nonnegative, $D(\nu, \nu) \geq 0$.
c) Use the above to compute
$$
\inf \left\{D(\rho, \rho): \rho(x) \geq 0, \int \rho=1, \rho(x)=0 \text { for }|x|>R\right\} .
$$
Is the infimum attained?