12 problems

For a metric space $M$ and a Banach space $Z$, define the Hahn–Banach extension operator constants … where … and $\operatorname{Ext}(A,M,Z)$ is the set of linear extension operator…

Let $n,m\geq 1$, let $\mu$ be a $\mathbb{R}^m$-valued Borel measure on $\mathbb{R}^n$ satisfying $\mu(\mathbb{R}^n)=0$, and consider the variational problem … A map…

Let $\mathbb{B}^n$ be the unit ball, let $\overline{\mathbb{R}}^n$ be the one-point compactification of $\mathbb{R}^n$, and let $q$ denote the chordal metric. For…

Let $X$ and $Y$ be sufficiently nice compact metric spaces, and let $L$ denote a Lipschitz bound for maps from $X$ to $Y$. The growth of homotopy classes represented by maps with L…

Let $\mu$ be an $\mathbb{R}^m$-valued Borel measure on $\mathbb{R}^n$ such that $\mu(\mathbb{R}^n)=0$ and $\mu$ is absolutely continuous with respect to Lebesgue measure. Let…

Let $Delta$ and $Delta'$ be planar polygons with corresponding vertices $xi,wj,uk,vl$ and $xi',wj',uk',vl'$, and let a subpolygon comparable to $Delta$ mean a subpolygon…

Let $Delta$ and $Delta'$ be planar polygons with three strictly convex vertices $x1,x2,x3$ and $x1',x2',x3'$, respectively, and let corresponding vertices be defined as in the pr…

Gromov's thickness conjecture. There is a homotopy between $f$ and $g$ of thickness $O(L^p)$, where $p$ depends only on the rational homotopy type of $Y$; perhaps $p$ can always be…

Let $F_j:M_j\to Y$ be $1$-Lipschitz, where $Y$ is compact, and suppose the $M_j$ satisfy Wenger's compactness hypotheses. After passing to a subsequence, assume … and that the push…

Let $f=(f_1,\ldots,f_m):\mathbb{Z}_k^n\to\mathbb{Z}_l^m$ be $\alpha$-Lipschitz with respect to the $L^1$ norm. Coordinatewise junta conjecture. For every $\delta,\epsilon>0$, at le…

Let $G=\mathrm{PO}(n,1)$, let $\Gamma_0$ be the group in the representation space under consideration, and let $(j,\rho)\in\operatorname{Hom}(\Gamma_0,G)^2$. For $n=2$, assume that…

Let $Y$ be an $m$-dimensional Alexandrov space with curvature bounded below by $-1$, and let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below by $-1$. Suppos…