8 problems

Nine dual spaces conjecture. There are precisely nine isomorphism types of dual Banach spaces arising from Type I AF C-algebras.

Let $K\subset\mathbf R^n$ be a symmetric convex body, let $\mathrm{GL}(n)$ denote the group of invertible linear maps on $\mathbf R^n$, and let $M$ and $M^\ast$ be the mean width a…

Let $p$ and $q$ satisfy $p\le 2\le q$, and let $A=(a_{ij})$ be a deterministic matrix. Let $(\varepsilon_{ij})$ be independent symmetric Bernoulli random variables, taking the valu…

Let $p$ and $q$ satisfy $p\le 2\le q$, and let $A=(a_{ij})_{i\le m,j\le n}$ be a deterministic $m\times n$ matrix. Write $p^*$ for the Hölder conjugate of $p$, and let…

Let $(\Phi,\Psi)$ be an admissible support function pair on a nest $\mathcal{E}$, and let $\mathcal M^0(\Phi,\Psi)$ be the bimodule constructed from this pair. Minimality conjectur…

The UMD$^-_{\rm PW}$ conjecture. If $\Phi\in\Delta_2$, then $L^{\Phi}(T)$ is UMD$^-_{\rm PW}$.

Let $\mathbb H$ be the discrete Heisenberg group, let $B_n$ be its radius-$n$ ball with the word metric $d_W$, and let $c_1(B_n,d_W)$ denote the least distortion of an embedding of…

Let $X$ be a finite-dimensional normed space, and let $m(X)$ denote the smallest possible cardinality of an equilateral subset of $X$ that is maximal with respect to inclusion. Swa…