8 problems

Let $X$ be a real Banach space with $\dim X>1$. The Daugavet property is the condition that $\|I+S\|=1+\|S\|$ for every rank-one operator $S\colon X\to X$. The square-Daugavet prob…

For each $q\ge 2$, does there exist a universal constant $C_q<\infty$ such that every bounded linear operator $T:X\to Y$ between Banach spaces satisfies … where…

Let $E$ be a Banach space, and let $(y_n^*)_{n=1}^{\infty}\subset S_{E^*}$ be a normal sequence, meaning that $y_n^*(x)\to 0$ for every $x\in E$. Must $(y_n^*)$ have a subsequence…

For $1\le p<\infty$, even $m\ge 4$, and $n\in\mathbb N$, let $A_p(n,m)$ be the least constant such that every function $f:\mathbb Z_m^n\to L_1$ satisfies … Determine the optimal or…

Let $\mathcal{A}$ be the class of complete Alexandrov spaces of nonnegative curvature. For integers $n\ge 1$ and dyadic $m\ge 2$, define $\Gamma_{\mathcal{A}}(n,m)$ to be the least…

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…

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…