辛凸与辛星形曲线--一个变分问题
Symplectically convex and symplectically star-shaped curves -- a variational problem
摘要：
在这篇文章中，我们提出了一个推广的2维概念的凸性。星形的辛向量空间。我们分别称这样的曲线为辛凸曲线。合星状在给出一些基本结果之后，我们研究了一类由仿射等周不等式所引起的辛凸和辛星形曲线的变分问题.这些变分问题可以还原为二维。对于一系列的族参数极值点的变分问题是刚性的：他们是多重遍历二次曲线。对于所有的族参数，我们确定当非平凡的一阶和二阶变形的二次曲线存在。在最后一节中，我们提出了一些示意图和问题，并在Gil Bor的Mathematica applet的帮助下创建了两个画廊。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
