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| 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- ω||)≤ f(||y- ω||)+ f(||w-k||)+Cf(||z-k||)+Cf(||y-z||).
并且给出了该结论在L_p空间和非交换的L_p空间以及某些内插空间上的应用.这一工作是 Enfo 关于圆度不等式及 Sch?tz 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:国家自然科学基金项目（面上项目，重点项目，重大项目） |
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| Geometric Properties of Banach Spaces and the Quadruple Inequality for Convex Functionals |
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Xie Zixiu
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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. Specifically, we explore the monotonicity and concavity-convexity of the convex function f under specifc conditions. For a p-uniformly smooth Banach space with 1 < p< 2, we establish a quadruple inequalityPrecisely, for any y, z, k, w in X, the following inequality holds:
f(||y- k||) + f(||z- ω||) ≤ f(||y- ω||)+ f(||ω-k||)+Cf(||z-k||)+Cf(||y-z||).
Furthermore, we present the applications of this conclusion in L_p spaces, non - commutative L_p spaces, and certain interpolation spaces. This research represents a generalization of the roundness inequality and Sch?tz'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. |
