12 problems

Furstenberg's countability conjecture. The set $E(K)$ should be no more than countable.

Let $1<l<n$, let $E\subset\mathbb R^n$ be a Borel set, and let $G(n,l)$ denote the Grassmannian of $l$-dimensional linear subspaces of $\mathbb R^n$. For $V\in G(n,l)$, write…

Let $X\subseteq\mathbb{R}^n$ be an affine variety, let $K\subseteq\mathbb{R}^n$ be a generic linear subspace with $\dim K=k<n-d$, where $d=\dim X$, and let…

Let $\Pi\texttt{net}$ denote the method studied in this paper, let $\texttt{DC3}$ denote the comparison method, and let $\texttt{RS}$ denote the reported residual metric. The train…

Equality conjecture. For $n\geq2$, $0<a\leq n$, $1\leq k\leq n-1$, and $\max\{0,a-n+k\}<s\leq\min\{a,k\}$,

The exceptional-set projection dimension conjecture. One should have

Let $p$) be prime and let $E oindentsubset mathbb{F}p^2$ satisfy $|E|=p^a$, where $a/2le sle ale 1$. For $Ts^{1,2}(E)$, the set of one-dimensional subspaces whose orthogonal projec…

Let $K\subset\mathbb R^3$ be a convex body and let $p$ be a point in its interior. For $u\in\mathbb S^2$, let $\pi_u$ denote orthogonal projection in direction $u$. Point equichord…

Let $K,L\subset\mathbb R^3$ be convex bodies with $L\subset\operatorname{int}K$, and suppose that $L$ is strictly convex. For $u,w\in\mathbb S^2$ with $u\bot w$, let $\pi_u$ denote…

Let $Gr(m,\mathbb{R}^n)$ be the Grassmannian of $m$-dimensional linear subspaces, let $\Lambda\subset Gr(m,\mathbb{R}^n)$, and let $E\subset\mathbb{R}^n$ be a compact Borel set. Fo…

Fix $0\leq t/2\leq s\leq t\leq 1$, and let $K\subset\mathbb{R}^2$ be a Borel set with $\dim_{\mathrm{H}}K\geq t$. For $e\in S^1$, let $\pi_e:\mathbb{R}^2\to\mathbb{R}$ be the ortho…

Let $K\subset\mathbb{R}^{2}$ be a Borel set, and let $\pi_e\colon\mathbb{R}^{2}\to\mathbb{R}$ be the orthogonal projection $\pi_e(x)=x\cdot e$ for $e\in S^{1}$. Assume that…