Starrheit der Extremwerte der \(L_p\)-Norm der zentroaffinen Krümmung in zentroaffiner Parametrisierung

In their paper  P. Albers and S. Tabachnikov proposed a generalization of the $2$-dimensional notions of convex and star-shaped curves, calling them symplectically convex and symplectically star-shaped.

By giving such curves a fixed parametrization called centroaffine parametrization, one finds that the curve is determined by its initial condition and its so called centroaffine curvature.

Similiar to Watanabe we want to now study the $L_p$ norm of this centroaffine curvature. We find that ellipses are always extremes of this functional. For some parameters there also exist other extremes.We then determine when first and second order deformations of ellipses exist.