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 sym...
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| Main Authors: | , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
2020
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| In: |
Arxiv
Year: 2020, Pages: 1-23 |
| DOI: | 10.48550/arXiv.2012.14797 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2012.14797 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2012.14797 
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| Author Notes: | Peter Albers, Serge Tabachnikov |
MARC
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| 520 | |a 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. | ||
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