\begin{align}
    r_{\text{opt}}&= \left[ \dfrac{4 (2\sqrt{\pi})^{d} \left(3 \Gamma\left(d + 3\right) \Gamma\left(2\right)
    \right)^2}{\tau_{p, d} (d+2)\left(\Gamma\left(d + 1\right) \Gamma\left(4\right)\right)^2}\right]^{1/(d+4)} 
     \cdot \hat{s} \cdot n^{-1/(d+4)} \nonumber\\ 
      &= \left((\sqrt{\pi})^d (d+1) \left(d+2\right) ! \right)^{1/(d+4)}
     \cdot \hat{s} \cdot n^{-1/(d+4)} \label{eq:app_l1expr}
\end{align}
