Carathéodory functions in the Banach space setting

Approximate Carath{é}odory's theorem in uniformly smooth Banach spaces

We study the 'no-dimension' analogue of Carath{é}odory's theorem in Banach spaces. We prove such a result together with its colorful version for uniformly smooth Banach spaces. It follows that uniform smoothness leads to a greedy de-randomization of Maurey's classical lemma \cite{pisier1980remarques}, which is itself a 'no-dimension' analogue of Carath{é}odory's theorem with a probabilistic proof. We find the asymptotically tight upper bound on the deviation of the convex hull from the $k$-convex hull of a bounded set in $L_p$ with $1 < p \leq 2$ and get asymptotically the same bound as in Mau…

The theorems of Caratheodory and Gluskin for $0<p<1$

In this note we investigate some aspects of the local structure of finite dimensional $p$-Banach spaces. The well known theorem of Gluskin gives a sharp lower bound of the diameter of the Minkowski compactum. In [Gl] it is proved that diam$({\cal M}_n^1)\geq cn$ for some absolute constant $c$. Our purpose is to study this problem in the $p$-convex setting. In [Pe], T. Peck gave an upper bound of the diameter of ${\cal M}_n^p$, the class of all $n$-dimensional $p$-normed spaces, namely, diam$({\cal M}_n^p)\leq n^{2/p-1}$. We will show that such bound is optimum.

Darbo-type theorem for quasimeasure of noncompactness

The paper introduces the concept of quasimeasure of noncompactness. Motivated by the Arzelà-Ascoli theorem for $C^b(X,E)$, where $X$ is an Euclidean space and $E$ an arbitrary Banach space, we construct a quasimeasure for this space and study its properties. An analogon for Darbo fixed point theorem is obtained with the additional aid of measure of nonconvexity. The paper ends with possible application in integral equations. We prove that a Hammerstein operator with Carathéodory kernel and nonlinearity of a certain type has a fixed point.

非拟连续牛顿函数与基于Banach函数空间的外容量

Non-quasicontinuous Newtonian functions and outer capacities based on Banach function spaces

本文构造了在度量空间中基于拟Banach函数格定义的Sobolev型函数，证明其缺乏拟连续性，与$L^p$情形形成对比，并探讨了Sobolev容量作为外容量的性质，指出Vitali--Carathéodory性质的缺失是关键。

Fixed points and normal automorphisms of the unit ball of bounded operators on $\mathbb{C}^n$

We examine the group of isometries of the open unit ball of a complex Banach space of certain bounded linear operators equipped with the Carathéodory metric. Therein we obtain a charactrization of the normal isometries in terms of their special type of fixed points.

关于有限度量空间上Lipschitz单位球的极值点与表示定理

On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces

本文刻画了有限度量空间到严格凸Banach空间的Lipschitz函数单位球的极值点，并提出表示定理，其所需极值点数量不依赖维度，优于Carathéodory定理。