12 problems

Given a Banach space $X$, let $D(X)$ be the diameter, in the Banach–Mazur distance, of the class of all Banach spaces isomorphic to $X$: … where $d(X_1,X_2)$ is the infimum of…

The complex cross-2 distance conjecture. For every integer $n\geq 1$ and all $1\leq q\leq 2\leq p\leq\infty$,

For $n\geq 1$, let $\ell_1^n(\mathbb{C})$ and $\ell_{\infty}^n(\mathbb{C})$ denote the $n$-dimensional complex sequence spaces, and let $d_{BM}$ denote the Banach–Mazur distance be…

For each integer $n\geq 2$, let $\mathcal{C}_n$ be the $n$-dimensional cube, let $\mathcal{C}_n^*$ be the $n$-dimensional cross-polytope, and let $d_{BM}$ denote the Banach–Mazur d…

Let $\mathcal{C}_3$ be the cube and $\mathcal{C}_3^*$ the cross-polytope in $\mathbb{R}^3$, and let $d_{BM}$ denote the Banach–Mazur distance between convex bodies. Let…

Distance conjecture. The upper estimates obtained for these distances are exact:

Let the Banach–Mazur compactum in dimension $n$ be the collection of $n$-dimensional Banach spaces modulo linear isomorphism, equipped with the weak distance. Rudelson's conjecture…

Let $X$ and $Y$ be $n$-dimensional Banach spaces, and let $\operatorname{wf}(X,Y)$ denote the weak Banach–Mazur distance quantity used in the paper. Rudelson's conjecture. There is…

Let the weak Banach–Mazur compactum be the space of $n$-dimensional Banach spaces equipped with the weak Banach–Mazur distance, and let its radius denote the corresponding radius.…

For each dimension $n$ and every $x\in\mathbb{R}^n$, define the average absolute inner product over the sign vectors $v_i\in\{-1,1\}^n$ by … The largest absolute constant $\alpha$…

Let $C_3=[-1,1]^3$ be the three-dimensional cube and let $C_3^\star$ be the three-dimensional crosspolytope. The exact Banach–Mazur distance conjecture. Based on computational resu…

Let $K$ be a convex body, and obtain bodies from $K$ by taking a section followed by a projection. Approximation is measured by Banach–Mazur distance, and “bounded by a constant” m…