12 problems

Let $G\subseteq\mathbb{R}^n$ be a centrally symmetric convex body. For $p\in(1,\infty]$, let $C_p(n,G)$ denote the best constant such that … where $\mathbf M_t^G$ is the integral H…

Guth–Zahl conjecture. Let $p\geqslant\frac{n}{n-1}$. For every $\varepsilon>0$, there are a complexity $D=D_{\varepsilon,n}\in\mathbb{N}$ and a constant $C_{\varepsilon,n}>0$ such…

Optimal-constant conjecture. The constant $\frac12$ in this inequality is not optimal; equivalently, there should be a universal constant greater than $\frac12$ for which the inequ…

Let $(X_i)_{i=1}^n$ be independent observations with law $P$, let $P_n$ be their empirical measure, and define the empirical process by $\mathbb{G}_n(f)=\sqrt{n}(P_n-P)(f)$. Let…

Let $(\mathcal H_t)_{t>0}$ be the Ornstein–Uhlenbeck transition semigroup on $\mathbb R^n$ with invariant measure $\gamma_\infty$. For $f\in L^1(\gamma_\infty)$, write…

Szatzschneider's conjecture. Under these conditions,

Let $G\subset\mathbb{R}^d$ be a bounded, closed, symmetric convex body with non-empty interior. For $t>0$, define … where $G_t=\{y\in\mathbb{R}^d:t^{-1}y\in G\}$ and…

Generic chaining conjecture. The same issue is conjectured to arise for the generic chaining argument by Talagrand (2005).

Divergence conjecture. There exists a function $f\geq 0$ such that

Let $F$ be a finitely generated free group, and let $w\mapsto T_w$ be an ergodic action of $F$ on a probability space $(X,\mathscr B,\mu)$. For $n\geq1$, let…

Let $(X,d,\mu)$ be a metric measure space with maximal operator $M$, and suppose that the measure of balls is independent of their centers. For $1<p\leq\infty$, strong type $(p,p)$…

Return Times conjecture. For every function $g\in L^q(X)$ there is a universal set $X_0\subseteq X$ with $\mu(X_0)=1$ such that, for every second dynamical system…