9 problems

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces extends to a surjective linear or conjugate-linear isometry between the space…

Step-isometry extension conjecture. The map $T$ can be extended to a step-isometry on the whole of $X$.

Isometric transitivity conjecture. If $u\in V$ with $\lVert u\rVert=1$ is a norming order unit, then there exists a surjective linear isometry

Let $d53d$ be a field, let $S_{n,d}(d53d)$ denote the space of homogeneous degree-$d$ polynomials in $n$ variables, and let $\operatorname{d}_{\operatorname{ess}}$ be the essential…

Let $(X,d)$ be a complete Riemannian manifold, where $d$ is its metric, and let $f:X\rightarrow X$ be a function. For $r>0$, the level set of the distance function at radius $r$ is…

Let $\Gamma$ be an infinite set with the discrete topology. The Banach space $\ell_{\infty}(\Gamma)$ consists of all bounded complex-valued functions on $\Gamma$, equipped with the…

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Let the considered two-dimensional manifolds be $\mathrm{CAT}(k)$ 2-manifolds in the sense of Gromov, and consider their isometries and fixed points. Fixed-point classification con…

Let ${\mathcal P}\subset\mathbb R^2$ be a finite point set. An isometry is a distance-preserving map $\varphi:\mathbb R^2\to\mathbb R^2$. The isometry overlap conjecture. There is…