賦距空間上凸性結構的討論

Some Discussions on Convexity in Metric Spaces

李世芳(Shih-Fang Lee)

指導教授 ： 李金城

中原大學/理學院/應用數學研究所/碩士(2017年)

https://doi.org/10.6840/cycu201700351

摘要

在傳統上凸分析是在實向量空間上來討論的，現在我們把它推廣到賦距空間上來討論。
在這篇論文中，我們首先在賦距空間上定義了凸集合和凸函數。我們檢視了一些範例，這些範例中的每個集合皆為凸集合。為避免此類顯而易見的討論，我們對賦距空間增加一些有意義的條件。
我們陳述並證明了[8]中的一些結果，最後討論了乘積賦距空間上的一些結果並加以證明。

關鍵字

賦距空間；凸性結構

並列摘要

Convex analysis is traditionally discussed in real vector spaces, now we generalize the concepts of convex sets and convex functions on metric spaces.
In this paper, we give definitions of d-convex sets and d-convex functions on metric spaces. We then inspect some examples in which every set is a d-convex set. To avoid such trivial cases, we propose a property for the metric spaces.
We state and prove some results in [8]. We finally try to make some discussions and prove some further results in product metric spaces.

並列關鍵字

Convexity in Metric Spaces

參考文獻

[8] Dongxian Lin(2007), Convexity in Metric Spaces, Master thesis, Department of Applied Mathematics, Chung Yuan Christian University

[1] N. Aronszajn, P. Panitchpakdi(1956), Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. Vol. 6, 405-439.

[3] Lai,H.C., Liu,J.C., Lee,K.E.S. (1999), Necessary and sufficient conditions for minimax fractional programming, J. Math. Ana. Appl., Vol. 230(2), pp. 311-328

[4] Luenberger, David G. (1969), Optimization by vector space methods, New York, Wiley.

[5] Rockafellar, R. Tyrrell (1970), Convex analysis, Princeton, N.J., Princeton University Press.