论文标题
在符号上凸和符合性星形曲线 - 一个变化问题
Symplectically convex and symplectically star-shaped curves -- a variational problem
论文作者
论文摘要
在本文中，我们提出了对凸的二维概念的概括。是星形到符号矢量空间。我们称此类曲线在凸面上称为resp。符合性星形。在提出了一些基本结果之后，我们研究了一个因素凸和恒星形曲线的变分问题家族，这是由仿射等级不平等的动机。这些变化问题可以简化回两个维度。对于各种问题的家族参数极端点的范围是刚性的：它们是跨越的锥体。对于所有家庭参数，我们确定圆锥形的一阶和二阶变形何时存在。在最后一部分中，我们提出了一些猜想和问题，以及在吉尔·博尔（Gil Bor）的Mathematica applet的帮助下创建的两个画廊。
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.
