利用距离函数的可微性证明Chebyshev集的凸性

希尔伯特空间中Chebyshev集的凸性

Convexity of Chebyshev Sets in Hilbert Spaces

本文研究了希尔伯特空间中切比雪夫集的凸性条件。

集合的凸性与其支撑函数的可微性的关系

Relations between the convexity of a set and the differentiability of its support function

在有限维空间中，具有非空内部的紧凸集的支撑函数除原点外可微当且仅当该集是严格凸的。本文对定义域内部的支撑函数的可微性与集合的凸性之间的关系进行了深入的研究，主要是对无界集合。然后，我们回顾了一些结果有关的可微性的费用函数与生产函数。

Differentiability of Distance Function and The Proximinal Condition implying Convexity

We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles [16]. Hence, our result advances that of Giles [16]. We prove that the proximinal condition of Giles [16] is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A n…

Smulian范数Frechet可微定理的若干几何结果

A Review on Some Geometric Results of the Smulian s Theorem on Frechet Differentiability of Norms

本文证明了Smulian关于范数的Frechet可微性的定理，并给出了它关于范数的Gateaux和Frechet可微性的一些几何结果，以及有关空间及其对偶的性质，如自反性和严格凸性.

A Note on the Chebyshev Set Problem in Normed Linear Spaces

Best approximation (BA) is an interesting field in functional analysis that has attracted a lot of attention from many researchers for a very long period of time up-to-date. Of greatest consideration is the characterization of the Chebyshev set (CS) which is a subset of a normed linear space (NLS) which contains unique BAs. However, a fundamental question remains unsolved to-date regarding the convexity of the CS in infinite NLS known as the CS problem. The question which has not been answered is: Is every CS in a NLS convex?. This question has not got any solution including the simplest form …

对数矩生成函数的高斯凸性及其在自旋玻璃中的应用

A Gaussian Convexity for Logarithmic Moment Generating Functions with Applications in Spin Glasses

本文证明了对数矩生成函数关于参数λ的凸性，并由此改进了高斯凸函数的下偏差不等式，给出了SK自旋玻璃模型中DFM猜想的定量界，并证明了负副本退火自由能的可微性。