On Asymmetric Distances
On Asymmetric Distances
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DOI：
10.2478/agms-2013-0004
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发表时间：
2013-06
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通讯作者：
A. Mennucci
中科院分区：
文献类型：
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作者：
A. Mennucci
摘要本文讨论了非对称长度结构和非对称度量空间。长度结构引出(半)距离函数；利用全变分公式，(半)距离函数引出长度。在第一部分中，我们在路径集中确定一个拓扑，该拓扑最好地描述了上述运算是幂等运算的情况。作为一个典型的应用，我们考虑由变分中的Finslerian泛函定义的路的长度。在第二部分中，我们推广了Busemann的一般度量空间的设置，并讨论了该理论的新发现：我们识别了三类有趣的路径，并对它们进行了比较；我们注意到测地线段(由Busemann定义)在我们的设置下不一定是连续的；因此，我们提出了三种不同的内在度量空间的概念。
Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.