Numerical verification of condition for approximately midconvex functions

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• 作者：P. Spurek (1) przemyslaw.spurek@ii.uj.edu.pl
  Ja. Tabor (1) jacek.tabor@ii.uj.edu.pl
• 关键词：Midconvex function – convexity – numerical verification
• 刊名：Aequationes Mathematicae
• 出版年：2012
• 出版时间：July 2012
• 年：2012
• 卷：83
• 期：3
• 页码：223-237
• 全文大小：317.5 KB
• 参考文献：1. Application to verify Condition T, http://www2.im.uj.edu.pl/badania/preprinty/imuj2011/pr1106.cpp
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• 作者单位：1. Institute of Computer Science, Jagiellonian University, ?ojasiewicza 6, 30-348 Kraków, Poland
• ISSN：1420-8903

文摘

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if f (\fracx + y2)-\fracf(x) + f(y)2 ￡ a(||x - y||) for x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.