  Recently, M.K.-H. Kiessling and A.S. Tahvildar-Zadeh proved that a unique
global classical solution to the relativistic Vlasov-Poisson system exists
whenever the positive, integrable initial datum is spherically symmetric,
compactly supported in momentum space, vanishes on characteristics with
vanishing angular momentum, and for $\beta \ge 3/2$ has
$\mathfrak{L}^{\beta}$-norm strictly below a positive, critical value
$\mathcal{C}_{\beta}$. Everything else being equal, data leading to finite time
blow-up can be found with $\mathfrak{L}^{\beta}$-norm surpassing
$\mathcal{C}_{\beta}$ for any $\beta >1$, with $\mathcal{C}_{\beta}>0$ if and
only if $\beta\geq 3/2$. In their paper, the critical value for $\beta =
{3}/{2}$ is calculated explicitly while the value for all other $\beta$ is
merely characterized as the infimum of a functional over an appropriate
function space. In this work, the existence of minimizers is established, and
the exact expression of $\mathcal{C}_{\beta}$ is calculated in terms of the
famous Lane-Emden functions. Numerical computations of the
$\mathcal{C}_{\beta}$ are presented along with some elementary asymptotics near
the critical exponent ${3}/{2}$.
