91 problems

Let $D\subset\mathbb{R}^2$ be a simply connected domain, and understand equality of domains modulo sets of measure zero. The Pompeiu property is the property characterized in the s…

Mixed-norm boundedness conjecture. If $\Sigma(Q_1,Q_2)$ is well-curved, then these necessary conditions are sufficient for the mixed-norm estimate, with the possible exception of t…

Let $V_{n,m}$ denote the Stiefel manifold of orthonormal $m$-frames in ${\mathbb R}^n$, and let the higher-rank analogue of the cosine transform be the cosine transform on…

Let $V$ be an $n$-dimensional Euclidean vector space and let $G$ be a compact group acting transitively on the unit sphere of $V$. Write $Val_i^G(V)$ for the degree-$i$ component o…

Let $\Omega\subset\mathbb{R}^{N+1}$ be a bounded region whose boundary is homeomorphic to the unit sphere in $\mathbb{R}^{N+1}$. Williams conjecture. If $\partial\Omega$ is homeomo…

Let $(M,g)$ be a simple Riemannian manifold with boundary defining function $\rho$, and fix $\gamma>-1$. An $\alpha$-boundary defining function ($\alpha$-bdf) for $\partial_+SM$ is…

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 $S\subset\mathbb R^n$. A set of injectivity for the spherical means for $C_c(\mathbb R^2)$ is a set such that, for every $f$ in that class, $f\ast\mu_r(x)=0$ for all $r>0$ and…

Let $\omega$ be the symmetric measure on the sphere at infinity $S^2$ of hyperbolic space ${\mathbb H}^3$. For $f\in C^0_c({\mathbb H}^3)$, define the action … where $x$ is regarde…

Let $V$ be an $n$-dimensional Euclidean vector space, and let $V_1\in Val_1^\infty(V)$ denote the first intrinsic volume. For each integer $i$ with $0\leq i<n/2$, consider multipli…

Covering property conjecture. For any $n>0$ and $1\leq k\leq n-1$, there exists a non-negative measure $\mu\in\mathcal{M}(G(n,n-k))$ supported in $Z$ such that

Projection-diameter conjecture. There is some orthogonal projection to $\mathbb R^{n-1}$ in which the image of $\gamma$ has diameter at most

Let $P=A_1A_2A_3A_4A_5$ be a simple pentagon with … where indices are taken modulo $5$. The kernel of $P$, denoted $\operatorname{ker}(P)$, is the set of points $q\in P$ such that…

Let $Q=ABCD$ be a simple quadrilateral with … and suppose that $Q$ has an interior diagonal of integer length. Quadrilateral diagonal insertion conjecture. The open segment of that…

Let $G$ be a finite planar graph. An integral drawing of $G$ is a plane straight-line drawing $\varphi:V(G)\to\mathbb Z^2$ such that … for every edge $uv\in E(G)$. Harborth's conje…

Let $Z$ be the incidence submanifold of $Y\times X$ defining a double fibration transform, and let $n”$ denote the codimension of the transform. Levelset integral kernels may use a…

Crofton-type conjecture. There exist a constant $C>0$ and a smooth function $\rho:M\to\mathbb{R}_{>0}$, depending only on $\mathcal{D}$ and $L$ and satisfying

Let $D\subset\mathbb{R}^d$ be a domain whose boundary $\partial D$ is homeomorphic to the unit sphere in $\mathbb{R}^d$. Higher-dimensional Pompeiu conjecture. The domain $D$ has t…

Let $D\subset\mathbb{R}^2$ be a bounded, simply connected Lipschitz domain. A domain has the Pompeiu/Schiffer property if it is not among the exceptional domains for the associated…

Let $\widehat{\operatorname{Pol}}(n;1)$ be the space of confined equilateral $n$-gons, let $\phi_i$ be the $i$th turning angle, and let $\kappa$ denote total curvature. The limitin…

Let $(M,g)$) be a simple Riemannian manifold and let $f$ be a symmetric $m$-tensor field, written in local coordinates as … Define its geodesic ray transform by … where…

Let $(M,g)$ be a simple Riemannian manifold and let $f$ be a symmetric $2$-tensor field. Its geodesic ray transform is … where $\gamma:[0,l(\gamma)]\to M$ is a geodesic connecting…

Generalized support theorem. A similar support theorem should hold for arbitrary, not necessarily radial, functions.

Let $d\geq4$ be even and let $K\subset\mathbb{S}^d$ be a proper spherical convex body of class $\mathcal{C}^2_+$. Let $K^*$ be its spherical dual, let $\alpha_P(K)$ denote its peri…