论文标题
关于多样性和有限维度的Banach空间
On diversities and finite dimensional Banach spaces
论文作者
论文摘要
$ m $中的多样性$δ$是在$ [0，\ infty）$映射到$ m $的每组有限点上定义的函数，其属性是$δ（x）= 0 $ if in and仅当$ | | x | x | x | \ leq 1 $ and $Δ（x \ cup x \ cup yleqΔ（x \ cup z）+cup x $ cup y per z $ cup tin fin $ x，y，z \ subset m $带有$ | z | \ geq 1 $。它的重要性依赖于这样一个事实，即它们概括了度量距离的概念。 Our main contribution is the characterization of Banach-embeddable diversities $δ$ defined over $M$, $|M|=3$, i.e. when there exist points $p_i\in\mathbb R^n$, $i=1,2,3$, and a symmetric, convex, and compact set $C\subset\mathbb R^n$ such that $Δ（\ {x_ {i_1}，\ dots，x_ {i_m} \}）= r（\ {p_ {p_ {i_1}，\ dots，p_ {i_m} \}，c），c $ c $ c c $ c $ c $ c $ c $ c $ c $ c。
A diversity $δ$ in $M$ is a function defined over every finite set of points of $M$ mapped onto $[0,\infty)$, with the properties that $δ(X)=0$ if and only if $|X|\leq 1$ and $δ(X\cup Y)\leqδ(X\cup Z)+δ(Z\cup Y)$, for every finite sets $X,Y,Z\subset M$ with $|Z|\geq 1$. Its importance relies in the fact that, amongst others, they generalize the notion of metric distance. Our main contribution is the characterization of Banach-embeddable diversities $δ$ defined over $M$, $|M|=3$, i.e. when there exist points $p_i\in\mathbb R^n$, $i=1,2,3$, and a symmetric, convex, and compact set $C\subset\mathbb R^n$ such that $δ(\{x_{i_1},\dots,x_{i_m}\})=R(\{p_{i_1},\dots,p_{i_m}\},C)$, where $R(X,C)$ denotes the circumradius of $X$ with respect to $C$.
