iterations.[]{data-label="The iterations"}](plotTnu1_gray.eps "fig:"){width="\wid"} ![The first four iterations.[]{data-label="The iterations"}](plotTnu2_gray.eps "fig:"){width="\wid"} ![The first four iterations.[]{data-label="The iterations"}](plotTnu3_gray.eps "fig:"){width="\wid"} ![The first four iterations.[]{data-label="The iterations"}](plotTnu4_gray.eps "fig:"){width="\wid"}

Clearly the $T$ iterations are much slower in converging to a round sphere. Not until the 3rd iteration does it become convex. At the other extreme lie the $T_{\nu}$ iterations where the first iteration is already almost indistinguishable from a round sphere. Intermediate between the two are the $T_K$ iterations. This visually depicts the the observations in section \[subsection: Asymptotic behavior\], where rates of convergence were compared using asymptotic behavior.

Higher dimensional projective space {#section: Higher dimensional projective space}
===================================

Let us now investigate the complex projective space $X = {\mathbb{CP}}^n$ where $n>1$. We will consider exclusively the ${T_{\nu}}$ iteration. Let $z_1, \ldots, z_n$ be local coordinates on $X={\mathbb{CP}}^n$. Let us fix once and for all a volume form $\nu$ on $X$ by using that induced by the normalized Fubini-Study metric. That is if $$\begin{gathered}
    \omega = \frac{\sqrt{-1}}{2\pi}\ \partial \overline{\partial}\, \log \left( 1+ \sum | z_k |^2 \right)
    \phantom{XXXXX} \\[3mm]
        \phantom{XXXXX} =
        \frac{\sqrt{-1}}{2\pi}\
        \frac{
                \sum_{i,j} \left[ \left( 1 + \sum_k |z_k|^2 \right) \delta_{ij} - z_j \overline{z}_i \right]
                }{
                \left( 1 + \sum_k |z_k|^2 \right)^2
        }
        dz_i \wedge d\overline{z}_j\end{gathered}$$ is 