We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say $x$ and $y$, which define it satisfy that the metric segment $[x, y]$ only contains $x$ and $y$. The most notable among them is the case when the free space admits an isometric predual with some additional properties. As an application, we get some new consequences about norm-attainment in spaces of vector valued Lipschitz functions.
Extremal Selections of Multifunctions Generating a Continuous Flow
设$F:[0,T]\times\R^n\mapsto 2^{\R^n}$是一个连续的多功能紧凑，不一定凸值。本文证明了，如果$F$满足以下Lipschitz选择性质：\begin{itemize} \item[{(LSP)}] {\sl For every $t,x$, every $y\in \overline{co} F(t,x)$ and $\varepsilon>0$, there exists a Lipschitz selection $ϕ$ of $\overline{co}F$, defined on a neighborhood of $(t,x)$, with $|ϕ(t,x)-y|<\varepsilon$.} \end{itemize}，则存在$ext F$的可测选择$f$，使得对于每个$x_0$，Cauchy问题$$ \dot x(t)=f(t,x(t)),\qquad\qquad x(0)=x_0 $$有唯一的Caratheodory解，且连续依赖于$x_0$ .我们注意到每个具有紧值的Lipschitz泛函满足（LSP）。另一个有趣的类（LSP）成立，由那些连续多元函数$F$组成，其值是紧的并且具有非空内部的凸闭包。
Bounded Extremal Problems in Bergman and Bergman-Vekua spaces
对Lipschitz光滑非零实值函数f和1 < p < $\infty$，本文分析了复平面单位圆上Vekua方程$\partial$ w =（$\partial$ f /f）w解的广义解析函数的Bergman空间A p f（D）.考虑Bergman空间A p（D）及其推广形式A p f（D）中的一族有界极值问题（最佳约束逼近），即通过对A p（D）或A p f（D）中的一个函数进行范数约束来逼近D的子集中的一个函数.初步建设性的结果提供了p = 2。
A Lipschitz spaces view of infinitely wide shallow neural networks
Institute of Applied Mathematics, TU Delft（代尔夫特理工大学应用数学研究所）；Department of Applied Mathematics, University of Twente（埃因霍温理工大学应用数学系）；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）；MaLGa - DIMA, University of Genoa（热那亚大学MaLGa-DIMA）；RoMaDS - Department of Mathematics, University of Rome Tor Vergata（罗马第二大学罗马达斯-数学系）
