Weak distances between random subproportional quotients of \(\ell^m_1\) (Q2903927)

The study of random quotients of finite-dimensional Banach spaces led to solutions of several problems in the local theory of Banach spaces. In fact, in the case of \(\ell_1^m\), random quotients exhibit extremal geometric properties within the class of Banach spaces of a given dimension.NEWLINENEWLINEIn this paper, the author studies lower bounds for weak distances between finite-dimensional Banach spaces of the same dimension. In particular, it is proved that the weak distance between random pairs of subproportional quotients of \(\ell_1^{n^2}\) is up to a logarithmic factor greater than the square root of the dimension. The main tools in the proofs are an argument due to Gluskin, used in the context of random quotients to solve the problem of Banach-Mazur distances, and a ``mixing operators'' argument, used by Szarek and Mankiewicz in solving the basis and symmetry constant problem. Since the methods previously yielded optimal estimates up to a logarithmic factor, the result suggests that the weak diameter of the Minkowski compactum of \(n\)-dimensional Banach spaces is of order \(\sqrt{n}\) (perhaps up to a logarithmic factor) as conjectured by Rudelson.NEWLINENEWLINEThe main result of Section 4 is the following: Given a \(\delta\)-rigid pair of \(n\)-dimensional Banach spaces \(X\) and \(Y\), the weak distance between them is up to an absolute constant greater than \(n\delta(\log n)^{-1}\), i.e. NEWLINE\[NEWLINE d_{w}(X,Y) \geq c n \frac{\delta}{\log n}. NEWLINE\]NEWLINE In Section 5, it is shown that the ``vast majority'' of pairs of random \(n\)-dimensional quotients of \(\ell_1^{n^2}\) is up to an absolute constant \(1/\sqrt{n\log n}\)-rigid. Using this as well as the main result of Section 4, the author obtains the main theorem, namely: \smallskip There are numerical constants \(c,c_0,C>0\) such that NEWLINE\[NEWLINE\mathbb P \times \mathbb P \left\{(B_1,B_2) \in \mathcal B_{n,n^2} \times \mathcal B_{n,n^2}:\, d_w(B_1,B_2) \geq c\sqrt{\frac{n}{\log^3n}} \right\} \geq 1-Ce^{-c_0n},NEWLINE\]NEWLINE where \(\mathcal B_{n,n^2}\) can be viewed as a space of random \(n\)-dimensional quotients of \(\ell_1^{n^2}\).