12 problems

Carnot tangent characterization. The following are equivalent:

A sub-Finsler Carnot group is a stratified, nilpotent Lie group equipped with a left-invariant distance. Its homogeneous dimension is denoted by $\mathcal{Q}$, and…

Quantitative Besicovitch conjecture. There exist positive constants $\varepsilon$ and $\delta$ such that, whenever $E\subset\mathbb R^2$ is compact and

Let $\mathbb G$ be a sub-Finsler Carnot group, and let its metric spheres be the level sets of the Carnot–Carathéodory distance associated with a sub-Finsler structure. Euclidean r…

Let $\mathbb G$ be a sub-Finsler Carnot group of step $2$, with Carnot–Carathéodory distance $d_{\mathrm{cc}}$, abnormal set $\mathrm{Abn}$, and a fixed Riemannian metric…

Let $s\in (0,1)$ and $C_0\in (1,\infty)$. Let $E\subset \mathbb{R}^2$ be a bounded AD-regular set with constant $C_0$, meaning that $E$ is closed and, for every $x\in E$ and…

Let $E\subset\mathbb{R}^2$ be $\mathcal{H}^1$ measurable and purely $1$-unrectifiable, with $\mathcal{H}^1(E)<\infty$. Besicovitch's almost-everywhere line intersection conjecture.…

Let $(X,\mathsf d,\mathcal{H}^{N})$ be a non-collapsed $\mathsf{RCD}(K,N)$ space. Let $\boldsymbol\partial^{*}X$ denote its reduced boundary. Boundary rectifiability conjecture. Th…

Let … -free measure, and let … … satisfies … This would improve the known lower bound involving … . The conjecture was also advanced by Raita and appears in related work cited by t…

Let $\mu$ be a locally finite Borel measure on $\mathbb{R}^n$, let $Q_1\in\vec{\Delta}(\mathbb{R}^n)$, and let $\lambda>1$ be the constant from Lerman's theorem. Define … Suppose t…

Cheeger's conjecture.

Let $\mathbb G$ be a Carnot group, let $E\subset\mathbb G$ be a set with constant horizontal normal $\nu\in\mathbf S^{m-1}$, and let $H$ be a vertical halfspace with the same horiz…