Pyramids in the complex projective plane (Q1179133)

For a metric space \(M\) and a natural number \(n\) the function \(\delta: M^ n\to\mathbb{R}\), \(\delta(x_ 1,\dots,x_ n)=\max_{ij}\hbox{dist}_ M(x_ i,x_ j)\) is called the diameter functional. In a previous paper the author constructed a suitable right inscribed pyramid on a \((2k+1)\)- gon whose set of vertices \(P_ k\) is a local minimum of \(\delta\) on \((S^ 2)^{2k+2}\). \(P_ k\) is embedded into \(\mathbb{C} P^ 2\) via \(P_ k\subset S^ 2=\mathbb{C} P^ 1\subset\mathbb{C} P^ 2\). It is shown that \(P_ k\) remains a local minimum of \(\delta\) on \((\mathbb{C} P^ 2)^{2k+2}\). Furthermore, an extremum of \(\delta\) on \((\mathbb{C} P^ 2)^ 6\) is constructed which is not contained in a totally geodesic \(\mathbb{C} P^ 1\).