27 problems

Let $\mu$ be an Ahlfors regular measure, and consider its Riesz transform. David–Semmes conjecture. The $L^{2}$-boundedness of the Riesz transform should be sufficient to imply rec…

Let $\mu$ be an arbitrary Radon measure in $\mathbb{R}^d$. A measure is $n$-rectifiable when there are Lipschitz maps $f_i:\mathbb{R}^n\to\mathbb{R}^d$, $i=1,2,\ldots$, such that t…

Let $E\subset\mathbb{R}^{2}$ be a $1$-set, meaning that $E$ has locally finite $\mathcal{H}^{1}$ measure, and let … Define $\sigma_{1,2}$ as the greatest value that the lower $1$-d…

Let a domain have separated connected components with harmonic measures defined on their common boundary. Bishop's two-phase conjecture. Mutual absolute continuity of the harmonic…

Let a domain have a boundary, with harmonic measure and surface measure defined on that boundary. Bishop's conjecture. Absolute continuity of harmonic measure with respect to surfa…

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…

Let $\omega:(0,1)\to(0,\infty)$ be a modulus of continuity, and consider the prescribed volume form equation … for a density function $f$ on $[0,1]^d$, where…

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

Besicovitch's conjecture. If $\Theta^1_*(E,x)>\frac{1}{2}$ for $\mathcal{H}^1$-almost every $x\in E$, then $E$ is countably $1$-rectifiable.

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 $E$ be a subset of a metric space, let $\mathscr{H}^\alpha$ denote the $e3e3alpha$-dimensional Hausdorff measure, and let $\mathscr{H}^\alpha\llcorner E$ be its restriction to…

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 $E\subset\mathbb{R}^2$ be $\mathcal{H}^1$ measurable, with $\mathcal{H}^1(E)<\infty$, and suppose that $E$ is purely $1$-unrectifiable. Besicovitch's one-half density conjectur…

Let $\Omega$ be the domain and let $u\in {\mathcal M}_{\rm div}(\Omega)$, with lifting $\phi$ and associated jump set $J$ and measure $\nu$. For functions in…

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…

The squared Wasserstein alpha-number rectifiability conjecture. The preceding rectifiability conclusion should remain valid when the integrand $\widetilde{\alpha}_{\mu}^{d}(x,r)$ i…

Besicovitch's conjecture. The constant $3/4$ in the second density condition may be replaced by $1/2$; that is,

Let $m$ be a positive integer, and let $n\in\{2,\dots,m-1\}$. Let $\boldsymbol{\mathsf{z}}_1,\dots,\boldsymbol{\mathsf{z}}_n$ be independent continuous random variables in…

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…

Let $\mu$ be a singular rectifiable measure, and let $\widetilde{J}_2(\mu,\cdot)$ denote its density-normalized weighted $L^2$ Jones function. Jones's conjecture. Weighted $L^2$ Jo…

Cheeger's conjecture.

Let $E\subset{\Bbb R}^d$, with $d\geq 2$, have Hausdorff dimension $d-1$. Suppose that $E$ is rectifiable and is not contained in a hyperplane. Rectifiable-set conjecture. Then the…