Computing Borel complexity of some geometrical properties in Banach spaces
Ginés López-Pérez, Esteban Martínez Vañó, Abraham Rueda Zoca
TL;DR
The paper advances the descriptive-set-theoretic analysis of geometric Banach-space properties by computing their Borel complexity within modern codifications. It shows isomorphism classes of spaces with diameter-2 properties, the Daugavet property, and octahedral norms are complete analytic in the standard seminorm coding, while isometry classes exhibit distinct and often higher-theory complexity, with Gurariĭ- and Gurariĭ-like universality playing key roles. In the $\mathcal{B}$-coding, diameter-2 properties and the Daugavet property are established as $G_\delta$-complete, and analogous results are obtained for the octahedral properties; in the infinite-dimensional $\mathcal{P}_\infty$ coding, many results yield $F_{\sigma\delta}$ upper bounds, with several properties shown to be $G_\delta$-complete via norm-based descriptions (e.g., LD$\delta$P) and dentability considerations. The work also develops dual-space–based and purely geometric methods to analyze these complexities, highlights partial optimality results, and raises open questions about exact classifications (notably for D2P, DD2P, and WOH) and the behavior of isometry classes in $\mathcal{P}_\infty$.
Abstract
We compute the Borel complexity of some classes of Banach spaces such as different versions of diameter two properties, spaces satisfying the Daugavet equation or spaces with an octahedral norm. In most of the above cases our computation is even optimal, which completes the research done during the last years around this topic for isomorphism classes of Banach spaces.