论文标题
几乎是不相关的家庭和不可分割的球的几何形状
Almost disjoint families and the geometry of nonseparable spheres
论文作者
论文摘要
我们考虑$ \ mathbb n $的子集的几乎不相关的家族，约翰逊 - 林斯特劳斯·巴纳克（Johnson-Lindenstrauss Banach Spaces）$（\ Mathcal x _ {\ Mathcal a}，\ | \ | \ | _ \ | _ \ infty $由它们引起\ | _ {\ infty，2}）$。我们引入了部分顺序$ \ Mathbb p _ {\ Mathcal a} $，并表征$（\ Mathcal x _ {\ Mathcal a}的球形的某些几何属性\ | _ {\ infty，2}）$在$ \ mathbb p _ {\ mathcal a} $的组合属性方面。利用一些已知和一些新的几乎不相交的家庭的极端行为，我们显示了Banach空间的存在，该单位球体显示出令人惊讶的几何形状： 1）有一个Banach密度连续体的空间，其单位球体是数量可计的，严格小于$ 1 $。 2）一致的是，每$ρ> 0 $都有一个不可分割的Banach空间，其中每$δ> 0 $都有$ \ varepsilon> 0 $，因此每个不可容纳的$（1- \ varepsilon）$（1- \ varepsilon）$ - 分离的单位元素的单位元素集合两个元素包含两个元素，而不是$ 1 $ $ 1 $ $ $ 2-至少$ 2--至少$ 2--至少$ 2--至少2-2--至少2-2-2--2--2-- 2-- 2-- 2--至今 应当指出的是，每一个$ \ varepsilon> 0 $每个不可分离的Banach空间都有大量不可数的$（1- \ varepsilon）$ - 由Riesz Lemma分开的集合。 我们还为表格$（\ Mathcal x _ {\ Mathcal a}，\ | \ \ | _ {\ iftty，2}）$的形式的空间获得一致的二分法：开放的彩色axiom意味着表格$（\ Mathcal x _ _ {\ Maths的每个BANACH空间\ | _ {\ infty，2}）$是可计数直径的结合，严格小于$ 1 $，或者它包含一个无数$（2- \ varepsilon）$ - 分离的$ \ varepsilon> 0 $。
We consider uncountable almost disjoint families of subsets of $\mathbb N$, the Johnson-Lindenstrauss Banach spaces $(\mathcal X_{\mathcal A}, \|\ \|_\infty)$ induced by them, and their natural equivalent renormings $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$. We introduce a partial order $\mathbb P_{\mathcal A}$ and characterize some geometric properties of the spheres of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty})$ and of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ in terms of combinatorial properties of $\mathbb P_{\mathcal A}$. Exploiting the extreme behavior of some known and some new almost disjoint families among others we show the existence of Banach spaces where the unit spheres display surprising geometry: 1) There is a Banach space of density continuum whose unit sphere is the union of countably many sets of diameters strictly less than $1$. 2) It is consistent that for every $ρ>0$ there is a nonseparable Banach space, where for every $δ>0$ there is $\varepsilon>0$ such that every uncountable $(1-\varepsilon)$-separated set of elements of the unit sphere contains two elements distant by less than $1$ and two elements distant at least by $2-ρ-δ$. It should be noted that for every $\varepsilon>0$ every nonseparable Banach space has a plenty of uncountable $(1-\varepsilon)$-separated sets by the Riesz Lemma. We also obtain a consistent dichotomy for the spaces of the form $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$: The Open Coloring Axiom implies that the unit sphere of every Banach space of the form $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ either is the union of countably many sets of diameter strictly less than $1$ or it contains an uncountable $(2-\varepsilon)$-separated set for every $\varepsilon>0$.
