由Transit Functions生成的凸几何
Convex Geometries yielded by Transit Functions
AI总结：
本文研究了由transit functions生成的凸几何，从公理化角度刻画了Minkowski-Krein-Milman性质，并将其应用于图上的若干著名transit functions。
摘要：
设$V$为有限非空集合。Transit function是一个映射$R:V\times V\rightarrow 2^V$，满足对于所有$u,v\in V$，$R(u,u)=\{u\}$，$R(u,v)=R(v,u)$且$u\in R(u,v)$。若对于$K$中的任意$u,v$都有$R(u,v)\subset K$，则集合$K\subseteq V$是$R$-凸的，且$V$的所有$R$-凸子集构成一个凸性$\mathcal{C}_R$。我们从公理化角度考虑Minkowski-Krein-Milman性质，即凸性$\mathcal{C}_R$中的每个$R$-凸集$K$均为$K$的极点集合的凸包，并给出了其刻画。随后我们考虑图上几个著名的transit functions，并展示了上述刻画在它们之上的应用。
英文摘要：
Let $V$ be a finite nonempty set. A transit function is a map $R:V\times V\rightarrow 2^V$ such that $R(u,u)=\{u\}$, $R(u,v)=R(v,u)$ and $u\in R(u,v)$ hold for every $u,v\in V$. A set $K\subseteq V$ is $R$-convex if $R(u,v)\subset K$ for every $u,v\in K$ and all $R$-convex subsets of $V$ form a convexity $\mathcal{C}_R$. We consider Minkowski-Krein-Milman property that every $R$-convex set $K$ in a convexity $\mathcal{C}_R$ is the convex hull of the set of extreme points of $K$ from axiomatic point of view and present a characterization of it. Later we consider several well-known transit functions on graphs and present the use of the mentioned characterizations on them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
