| 摘要: |
本文研究了Banach空间上的凸泛函四重不等式,该不等式联系于Banach空间中的凸泛函的几何性质及凸函数的光滑性条件.具体地,研究了凸函数f满足一定条件下的单调性和凹凸性,并在1≤p≤2时的p一致光滑Banach空间上,建立了四重不等式,即对任意y,z,k,w∈X,有
f (||y-k||)+f (||z-w||)≤ f (||y-w||)+f (||w-k||)+Cf (||z-k||)+Cf (||y-z||).
并且给出了该结论在Lp空间和非交换的Lp空间以及某些内插空间上的应用.这一工作是Enflo关于圆度不等式及Schötz在Hilbert空间上的凸泛函四重不等式在Banach空间框架下的推广. |
| 关键词: Banach空间几何 四重不等式 p一致光滑性 Clarkson不等式 凸泛函 |
| DOI： |
| 分类号:O177.2 |
| 基金项目:国家自然科学基金资助(12071358). |
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| GEOMETRIC PROPERTIES OF BANACH SPACES AND THE QUADRUPLE INEQUALITY FOR CONVEX FUNCTIONALS |
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XIE Zi-xiu, MA Tao
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School of Mathematics and Statistics, Central South Minzu University, Wuhan 430074, China
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| Abstract: |
In this paper, we delve into the quadruple inequality for convex functionals in Banach spaces. This inequality is intricately linked to the geometric characteristics of convex functionals within Banach spaces and the smoothness conditions of convex functions. Speciflcally, we explore the monotonicity and concavity-convexity of the convex function f under speciflc conditions. For a p-uniformly smooth Banach space with 1≤p≤2, we establish a quadruple inequality. Precisely, for any y,z,k,w∈X, the following inequality holds:
f (||y-k||)+f (||z-w||)≤ f (||y-w||)+f (||w-k||)+Cf (||z-k||)+Cf (||y-z||).
Furthermore, we present the applications of this conclusion in Lp spaces, non-commutative Lp spaces, and certain interpolation spaces. This research represents a generalization of the roundness inequality and Schötz’s quadruple inequality for convex functionals on Hilbert spaces. |
| Key words: The geometry of banach spaces quadruple inequality p-uniform smoothness Clarkson’s inequality Convex functional |
