弱稳定性与切片的二元凸组合
Weak stability and binary convex combinations of slices
AI总结：
该研究探讨巴拿赫空间单位球切片凸组合的弱开性等几何性质，证明$\mathrm{P1}^{(m)}$与$\mathrm{CWO}^{(m)}$等价，构造了具有特定性质的无限维巴拿赫空间及$c_0$上的等价范数，回答了文献中的若干问题。
AI中文摘要：
我们研究巴拿赫空间单位球切片的凸组合的弱开性及相关几何性质。证明对每个$m\geq2$，性质$\mathrm{P1}^{(m)}$与$\mathrm{CWO}^{(m)}$等价，其闭包变体亦然；因此$\mathrm{P1}$与$\overline{\mathrm{P1}}$均由两个切片的凸组合决定，特别地$\mathrm{P1}$与$\mathrm{CWO}$等价。构造了具有$\mathrm{P2}^{(2)}$但无$\mathrm{P2}$、具有$\mathrm{P3}^{(2)}$但无$\mathrm{P3}$的无限维实巴拿赫空间；在后者中，每个两个切片的凸组合都包含距离为2的两点，尽管该空间不满足强直径二性质。最后，给出实空间$c_0$上的等价范数，使得$\overline{\mathrm{P1}}$成立但$\mathrm{P1}$不成立，这些结果回答了文献中提出的若干问题。
英文摘要：
We study weak openness and related geometric properties of convex combinations of slices of the unit balls of Banach spaces. We prove that, for every $m\geq 2$, properties $\mathrm{P1}^{(m)}$ and $\mathrm{CWO}^{(m)}$ are equivalent, and likewise for their closure variants. Consequently, both $\mathrm{P1}$ and $\overline{\mathrm{P1}}$ are determined by convex combinations of two slices; in particular P1 and CWO are equivalent. We construct infinite-dimensional real Banach spaces with $\mathrm{P2}^{(2)}$ but without P2, and with $\mathrm{P3}^{(2)}$ but without P3. In the latter space, every convex combination of two slices contains two points at distance 2, although the space fails the strong diameter two property. Finally, we give an equivalent norm on the real space $c_0$ for which $\overline{\mathrm{P1}}$ holds, but P1 fails. These answers several questions raised in the literature.
