Peakless and monotone functions on G-spaces
Peakless and monotone functions on G-spaces
复制标题
G 空间上的无峰且单调函数
DOI：
10.21099/tkbjm/1496159666
复制
发表时间：
1983
影响因子：
0.7
通讯作者：
P. B.B
中科院分区：
文献类型：
--
作者：
Busemann H.;P. B.B
近年来，关于完备黎曼流形上的凸集和函数的一个广泛而有意义的理论已经产生。在沃尔特身上发现了一项很好的最新调查[14]。本文提供了非黎曼空间上函数的一个类似理论的基础，即包含光滑完备Finsler空间的$Gspace^{1)}$。主要的区别(与可能的不光滑性无关)是，在许多情况下，比凸性弱的无峰性(见下文)证明了获得与Riemann空间中的结果相对应的结果的适当概念。我们的方法具有一些令人惊讶的效果，就像它们起源于黎曼情形一样，它们通常比原始方法产生更强的结果，因为黎曼空间上的无峰函数不必是凸的。我们非常感谢地强调，我们已将本文的原始版本作为预印本发送给N.Innami。他不仅发现了一些不准确的地方，而且大大加强了我们的定理(22)和(25)，并允许我们把这些结果包含在本文中。我们将引入各种类型的无峰函数，其中一种我们称之为“几乎无峰”。这个概念与黎曼情形中的连续“测地拟凸”函数(“测地”经常被省略)不谋而合。尽管我们不喜欢改变公认的术语，但实际上这种改变是出于语义和数学原因迫使我们在这里进行的。对函数几乎无峰、无峰或凸的要求越来越严格。
In recent years an extensive and significant theory of convex sets and functions on complete Riemann manifolds has been created. A good and up-to-date survey is found in Walter [14]. The present paper provides the foundations of an analogous theory for functions on non-Riemannian spaces, namely $Gspaces^{1)}$ , which include the smooth complete Finsler spaces. The principal difference (unrelated to the possible absence of smoothness) is that in many cases peaklessness (see below) which is weaker than convexity, proves the adequate concept for obtaining results corresponding to those in Riemann spaces. Our methods have the somewhat suprising effect that applied to the Riemannian case from which they originated they often yield stronger results than the original ones because a peakless function on a Riemann space need not be convex. We gratefully emphasize that we sent the original version of this paper as a preprint to N. Innami. He not only discovered some inaccuracies, but strengthened our Theorems (22) and (25) materially and permitted us to include the results in the present paper. We are going to introduce various types of peakless functions, one of which we call ”nearly peakless”. The concept coincides with the continuous “geodesically quasiconvex” functions (”geodesically” is often omitted) in the Riemannian case. Although we dislike changing accepted terminology, the change is practically forced on us here both by semantic and mathematical reasons. Requiring that a function is nearly peakless, peakless or convex are increasingly stringent con-