37 problems

Interior problem conjecture. There exists a function $f\in C_c^{\infty}(\overline D)$, not identically zero in $K$, such that $\mathcal R f(L)$ vanishes for every line $L$ that int…

Power-polynomial conjecture. Then $\partial K$ is an ellipsoid; consequently, one can take $m=1$ when $n$ is odd and $m=2$ when $n$ is even.

Quadraticity conjecture. Every smooth polynomially integrable hypersurface is a strictly convex quadric in $\mathbb R^{2k+1}$.

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 $S\subset\mathbb{R}^n$ and let $C_c(\mathbb{R}^n)$ denote the compactly supported continuous functions on $\mathbb{R}^n$. A rigid motion of $\mathbb{R}^n$ is denoted by…

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

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 $\mathbb{F}$ be one of $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, let $p=\dim_{\mathbb{R}}\mathbb{F}$, and let $K\subset \mathbb{F}^n$ be a convex body with non-empty interio…

Gauge uniqueness conjecture. If

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…

Layered-scattering determination conjecture. The averages $\mathcal L^{\mathsf{av}}_{0\leq j\leq\lceil k/2\rceil}$ determine the intrinsic global invariants…

Let $K \subset \mathbb R^n$ be a compact body with $C^{\infty}$ boundary $\partial K$. For $\xi \in \mathbb R^n$ with $|\xi|=1$ and $t \in \mathbb R$, let … be the sectional volume…

Let $m=2k+1$, let $c_1,\ldots,c_m$ be specified weights, let $\gamma_1,\ldots,\gamma_m$ denote ray directions, and let $\mathcal{S}$ be the corresponding star transform. A configur…

Let $b\mathbb{CP}^n_\lambdad$ be the complex space form of constant holomorphic curvature $4\lambda$, with invariant valuation algebra…

Let $V_j$ be finite-dimensional real vector spaces equipped with smooth norms $F_j$, and let $e_j\colon M\to V_j$, $j=1,2$, be smooth immersions of a smooth manifold $M$. For a val…

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$, and let $\widehat M$ be its convex hull. For each $a\in M$, let $A_M^a(t)$ denote the $(n-1)$-dimension…

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 $w:\mathbb{R}\to\mathbb{R}$ be a $2\pi$-periodic function of class $C^2(\mathbb{R})$. Integral relation. The function $w$ satisfies … The relation is presented as a generalizat…

Generalized Busemann intersection inequality. More generally, for an arbitrary real number $m$ such that the integral on the right-hand side exists in the Lebesgue sense,

The sharp norm conjecture for the $(j,k)$-plane transform. One has

Let $i$ be a nonnegative integer, let $\mathbb{R}^{i+1}$ be Euclidean space, and let $\operatorname{Val}_i^{\infty}(\mathbb{R}^{i+1})$ denote the smooth valuations of degree $i$ on…

Let $T$ be a Gaussian eigenfunction on a compact manifold $\mathbb M$. For $u\in\mathbb R$, define the excursion set … For each Lipschitz–Killing curvature of $A_u(T,\mathbb M)$, c…

Let $X\subseteq\mathbb{R}^2$ be a compact set suitable for the perimeter and classical Euler characteristic to be defined. Here $|tX|$ denotes the magnitude of the dilation of $X$…

Let $\operatorname{MA}(\mathbb{R}^d)$ denote the class used in the source, and let $\mathcal N$ be the class of compact sets admitting decreasing smooth-domain approximations with…