论文标题
凸组平滑地图
Smooth maps on convex sets
论文作者
论文摘要
从凸组到笛卡尔空间的平滑地图有几个概念。这些概念中的一些是一致的，但并非所有人都这样做。我们在平面的凸子集上构建一个实值函数，该功能不会扩展到凸的任何开放型社区上的平滑函数，但是每个$ k $都扩展到凸面集的开放型社区上的$ c^k $功能。因此，在凸组上的平滑度的差异和西科斯基概念不合时间。我们表明，对于局部关闭的凸集，这些概念确实是一致的。借助凸组的平滑度的差异学概念，我们表明差异空间的类别与所谓的详尽陈代空间的类别是同构的。
There are several notions of a smooth map from a convex set to a cartesian space. Some of these notions coincide, but not all of them do. We construct a real-valued function on a convex subset of the plane that does not extend to a smooth function on any open neighbourhood of the convex set, but that for each $k$ extends to a $C^k$ function on an open neighbourhood of the convex set. It follows that the diffeological and Sikorski notions of smoothness on convex sets do not coincide. We show that, for a convex set that is locally closed, these notions do coincide. With the diffeological notion of smoothness for convex sets, we then show that the category of diffeological spaces is isomorphic to the category of so-called exhaustive Chen spaces.
