On the shape of the moduli of spherical minimal immersions

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold $M$ is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of $M$. In this paper we define and study a sequence of metric invariants $\sigma_m$, $m\geq 1$, associated to a compact convex body $\mathcal{L}$ with base point $\mathcal{O}$ in the interior of $\mathcal{L}$. The invariant $\sigma_m$ measures how lopsided $\mathcal{L}$ is in dimension $m$ with respect to $\mathcal{O}$. The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate $\sigma_m$ for convex polytopes.