Pyramids in the complex projective plane

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We study the critical points of the diameter functional δ on the n-fold Cartesian product of the complex projective plane C P 2 with the Fubini-Study metric. Such critical points arise in the calculation of a metric invariant called the filling radius, and are akin to the critical points of the distance function. We study a special family of such critical points, P k⊂C P 1⊂C P 2, k=1,2... We show that P k is a local minimum of δ by verifying the positivity of the Hessian of (a smooth approximation to) δ at P k. For this purpose, we use Shirokov's law of cosines and the holonomy of the normal bundle of C P 1⊂C P 2. We also exhibit a critical point of δ, given by a subset which is not contained in any totally geodesic submanifold of C P 2.