  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.
