18 problems

Boundary curvature-integral conjecture. One has

Naber's measure-convergence conjecture. As measures,

Let $M$ be a Riemannian manifold. A valuation $\mu$ is called angular when … where $\mathcal{A}(M)$ is the space of angular curvature measures and $\cdot$ denotes the Alesker produ…

Let $M$ be a Riemannian manifold. The space of angular curvature measures is denoted by $\mathcal{A}(M)$, and the Lipschitz–Killing algebra by $\mathcal{L}\mathcal{K}(M)$. Here a v…

Let $L$ be a critical link, and let its curvature measure be the measure describing the curvature of $L$ along the link. Curvature-measure absolute continuity conjecture. The curva…

Synthetic curvature-measure conjecture. There exists a locally finite curvature measure

Let the corresponding spaces of continuous translation invariant curvature measures be the spaces studied by Schuhmacher and Wannerer in the cited work. Schuhmacher–Wannerer conjec…

Consider a sufficiently regular sequence of convex bodies $(K_n)_{n\geq 1}$ such that $\dim K_n=n$ for every $n$ and … as $n\to\infty$. For each $n$, let…

Let $n$ be a positive integer, let $\operatorname{Curv}_k^\pm(\mathbb{R}^n)$ be the even or odd degree-$k$ curvature measures, and define … This is an invariant closed subspace und…

Let $n$ be a positive integer, let $\operatorname{Curv}_k(\mathbb{R}^n)$ be the degree-$k$ space of curvature measures with its topology induced by compact convergence, and let…

Let $\operatorname{Curv}(\mathbb{R}^n)$ be the space of translation invariant curvature measures, graded by degree and parity, and let a graded component mean one of the even or od…

Let $X_i\in\text{Alex}^n(-1)$ be a sequence of Alexandrov spaces converging to $X$ in the Gromov–Hausdorff sense without collapsing. Let $\mu_{X_i}$ and $\mu_X$ denote their synthe…

Let $X\in\text{Alex}^n(-1)$, and let … be the synthetic curvature measure, where ${\operatorname R}$, $\mathcal{K}$, and $\theta$ are the corresponding density functions. Local fin…

Let $\phi$ be the norm defining the anisotropic curvature measures $\mathcal{C}_m^\phi(C,\cdot)$ of a convex body $C\subseteq\mathbf{R}^{n+1}$. Anisotropic curvature-measure charac…

Let $F\subset\mathbb R^d$ be bounded, let $n=\dim\operatorname{aff}F\geq 1$, and for $l\in\{n,\ldots,d\}$ let $s_k^l(F)$ denote the $k$-th scaling exponent computed from the $l$-di…

Let $K\subset\mathbb R^d$ be a self-similar set satisfying the open set condition (OSC). Let $s_k(K)$ denote its $k$-th scaling exponent and let $\dim_M K$ denote its Minkowski dim…

Let $A,B$ be compact WDC sets with d.c. auras $f,g$, respectively, and let $\operatorname{nor}(f,0)$ and $\operatorname{nor}(g,0)$ denote their unit normal bundles. Vanishing-produ…

A compact self-similar set $F\subset\mathbb{R}^d$ has parallel sets … The curvature measures of $F_\varepsilon$ are defined for almost every sufficiently small $\varepsilon$ when t…