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

TL;DR

This paper extends classical convexity and star-shaped concepts to symplectic vector spaces, studies related variational problems, and characterizes extremal curves, including conics, through a blend of geometric analysis and computational tools.

Contribution

It introduces symplectically convex and star-shaped curves, analyzes associated variational problems, and characterizes extremal solutions, including conditions for deformations of conics.

Findings

Extremal points are multiply traversed conics for certain parameters.

Existence of non-trivial deformations of conics depends on parameter ranges.

Provides conjectures, questions, and visualizations related to symplectic convexity.

Abstract

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.