9 problems

Dyn–Farkhi conjecture. One has

Let $C$ be a convex figure of perimeter one, and consider approximation by inscribed $k$-gons in the Hausdorff metric. Popov's conjecture. The regular $(k+1)$-gon is the worst case…

Let $\mathcal{P}$ be a family of convex polygons in $\mathbb{R}^{2}$ that is closed under taking subpolygons, where a subpolygon is the convex hull of a subset of the polygon's ver…

Let $K\subset\mathbb{R}^2$ be a symmetric convex body with strictly convex $C^1$ boundary. For nonempty compact sets $A,B\subset\mathbb{R}^2$, let $\mathcal{R}_K(a,b)$ be the famil…

Let $(X,d)$ and $(Y,\rho)$ be ultrametric spaces, with completions $(\tilde{X},\tilde{d})$ and $(\tilde{Y},\tilde{\rho})$. Let $\bar{\mathbf{B}}_X^0$ and $\bar{\mathbf{B}}_Y^0$ den…

Let $(X,d)$ be an ultrametric space, and let $(\tilde{X},\tilde{d})$ be its completion. Write $\bar{\mathbf{B}}_X$ and $\bar{\mathbf{B}}_{\tilde{X}}$ for the balleans of closed bal…

Average Hausdorff distance conjecture. The condition that $\mathcal M_X$ be a smooth submanifold could be relaxed by considering an average Hausdorff distance, for example,

Fix a parameter pair $(c_0,c_1)\in\mathbb{C}^2$ and let ${\bf s}$ be a template. For each $n$, let $\overline{\bf s}^{n}$ denote the finite $n$-root of ${\bf s}$, and let…

Fix a parameter pair $(c_0,c_1)\in\mathbb{C}^2$ and let ${\bf s}$ be a template. For a template ${\bf t}=(t_k)$, its $k$-root is the finite sequence…