Symplectically convex and symplectically star-shaped curves: a variational problem

In this article, we propose a generalization of the 2-dimensional notions of convexity resp. being star-shaped to symplectic vector spaces. We call such curves symplectically convex resp. symplectically star-shaped. After presenting some basic results, we study a family of variational problems for symplectically convex and symplectically star-shaped curves which is motivated by the affine isoperimetric inequality. These variational problems can be reduced back to two dimensions. For a range of the family parameter, extremal points of the variational problem are rigid: they are multiply traversed conics. For all family parameters, we determine when non-trivial first- and second-order deformations of conics exist. In the last section, we present some conjectures and questions and two galleries created with the help of a Mathematica applet by Gil Bor.