as tabulated in the following table:

   $\sigma$   $k=2$    3      4      5
  ---------- ------- ------ ------ ------
    $n=2$     0.07    0.14   0.21   0.28
      3       0.05    0.11   0.17   0.22

One can check that approximate fractional equivalents to these numbers follow the pattern $$\label{eqn: sigma for general n in sym case}
    \sigma = \frac{(k-1)k}{(k+n+2)(k+n+3)}.$$ We should stress that when $n=1$ equation (\[eqn: sigma for general n in sym case\]), together with (\[eqn: sigma for Tv generic case\]), specializes to (\[eqn: sigma for CP1\]). This together with various experimental evidence leads the author to ask the following:

Let $G$ be a torus-invariant metric arising from a matrix on $H^0({\mathbb{CP}}^n, O_{{\mathbb{CP}}^n}(k))$, and let $B = {T_{\nu}}^{\infty}(G)$ be the limiting balanced metric under the ${T_{\nu}}$ iteration. Define $$\sigma_G(n,k) :=
        \lim_{r \rightarrow \infty}
            \frac{
                \mathrm{dist}({T_{\nu}}^{(r+1)}(G),\ B)
            }{
                \mathrm{dist}({T_{\nu}}^{(r)}(G),\ B)
            }.$$ Let us say that $G$ is *generally symmetric* if it is invariant under some fixed-point-free permutation of the homogeneous coordinates. Then do we have the general formula $$\label{eqn:  sigma for Tv for general n}
    \sigma_G(n,k) =
        \left\{ \begin{array}{cl}
            \frac{(k-1)k}{(k+n+1)(k+n+2)} &
                \mathrm{if\ }G\mathrm{\ is\ generally\ symmetric} \\
            \frac{k}{(k+2)} & \mathrm{otherwise}
        \end{array} \right. \quad ?$$

Example computation {#subsection:  Example computation}
-------------------

To illustrate a typical computation leading to some of 