关于H-凸集的${\mathbb H}$ -锥函数
On the ${\mathbb H}$-cone-functions for H-convex sets
摘要：
给定Heisenberg群${\mathbb H}$的一个紧致H凸子集$K$，原点$e$在它的内部，我们有兴趣找到一个齐次H凸函数$f$，使得$f(e)=0$和$f\bigl|_{\partial K}=1$ ;我们称这个函数$f$为顶点$e$和基$\partial K$的${\mathbb H}$ -锥函数。虽然在欧几里得框架中这个问题的等价版本有一个简单的解决方案，但在我们的上下文中，这个研究结果是相当纠缠的，这个问题可能是不可解决的。我们遵循的方法利用了Fenchel引入的凸族概念的扩展。我们提供了精确的，即使尴尬，$K$所需的条件，使$\partial K$是一个${\mathbb H}$锥函数的顶点$e.$的基础通过适当的就业这个条件，我们证明了两个有趣的绑定约束的形状的集合$K,$连同几个例子。
英文摘要：
Given a compact and H-convex subset $K$ of the Heisenberg group ${\mathbb H}$, with the origin $e$ in its interior, we are interested in finding a homogeneous H-convex function $f$ such that $f(e)=0$ and $f\bigl|_{\partial K}=1$; we will call this function $f$ the ${\mathbb H}$-cone-function of vertex $e$ and base $\partial K$. While the equivalent version of this problem in the Euclidean framework has an easy solution, in our context this investigation turns out to be quite entangled, and the problem can be unsolvable. The approach we follow makes use of an extension of the notion of convex family introduced by Fenchel. We provide the precise, even if awkward, condition required to $K$ so that $\partial K$ is the base of an ${\mathbb H}$-cone-function of vertex $e.$ Via a suitable employment of this condition, we prove two interesting binding constraints on the shape of the set $K,$ together with several examples.
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