41 problems

Conjecture 1 (Generalized Chang--Yang conjecture). For every integer $N\geq 1$ and every $\alpha\geq\frac{1}{2}$, let

Let $p>2$, let $\mathbb{D}$ be the unit disk, and let $f$ be analytic in $\mathbb{D}$. Denote by $\|f\|_{H^p}$ the Hardy-space norm and by $\|f\|_{D_{p/2}}$ the norm of the corresp…

Let ${\mathcal F}_j$ be the spin-$j$ function space on ${\mathbb S}^2={\mathbb C}\cup\{\infty\}$, and for $p\geq 1$ define the normalized norm by … Let $K(z,w)$ denote the coherent…

Let $n\ge1$ and $k\ge2$, and let $K_1,\ldots,K_k\subset\mathbb R^n$ be origin-symmetric convex bodies. Kalantzopoulos–Saroglou's conjecture. If … for all $x_i\in K_i$,…

Let $n\ge1$ and $k\ge2$. Let $\rho:\mathbb R\to\mathbb R_+$ be non-increasing, and let $f_1,\ldots,f_k:\mathbb R^n\to\mathbb R_+$ be even integrable functions. For … assume … for a…

Let $\gamma$ denote the standard Gaussian measure on $\mathbb R$. For $n,d\in\mathbb N$, let $\mathcal P_d(\mathbb R^n)$ be the space of polynomials on $\mathbb R^n$ of degree at m…

Let $K$ be a convex body in $\mathbb{R}^n$ and let $(\sigma_1,\dots,\sigma_m)$ be an $s$-cover of $\sigma\subset[n]$. For each $\tau\subset[n]$, let $H_\tau$ denote the coordi…

KLS conjecture. There exists a universal constant $\mathbf{C}_{\mathrm{univ}}>0$ such that

Let $m_u,m_w,e_u,e_w\in[0,1]$ satisfy $H_2(m_u)\geq e_u$ and $H_2(m_w)\geq e_w$. Let $\phi$ and $J$ be the functions defined in the paper. The explicit lower-bound conjecture. One…

For $(u,w)\in(0,1)^2$, define … with $\kappa(u,u)=0$. The kappa inequalities. The following two assertions hold: for $u,w\in(0,1/2]$, $\kappa(u,w)\leq\kappa(1-u,w)$; and the functi…

The candidate function conjecture. The function $\phi$ belongs to the class $\Psi$. If true, this would establish the most-informative Boolean function conjecture for balanced func…

Let $s\in(-\frac{1}{n},0)$ and let $g:\mathbb{R}^{n}\to\mathbb{R}_{+}$ be even with $g^{s}\in\mathcal{F}$, where $\mathcal{F}$ is the set of convex lower semicontinuous functions…

Quantum KKL conjecture. There exists a universal constant $C>0$ such that for each $n\in\mathbb N$ and quantum Boolean function $T$,

Functional $L^p$-Mahler lower-bound conjecture.

Functional symmetric $L^p$-Mahler conjecture.

For $p>0$, let ${\mathbb{S}}^{n-1}_p=\{x\in\mathbb{R}^n:\|x\|_p=1\}$, where $\|x\|_p=(\sum_{i=1}^n|x_i|^p)^{1/p}$. Let $\mathbf{P}$ be the cone measure on ${\mathbb{S}}^{n-1}_p$, a…

Log-Sobolev KLS conjecture.

Let $d=1$ and $s\in(0,\frac{1}{2})$. For $f\in\dot{H}^s(\mathbb{R})\setminus\mathcal B$, let $\mathcal Q(f)$ denote the Bianchi–Egnell quotient, let $c_{BE}(s)$ be its infimum, let…

Let $\Omega$ be a finite unweighted graph and consider the stochastic matrix $P$ associated with the simple random walk on this graph, $\tilde{P} = \frac{P+I}{2}$ associated to the…

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$. It is $\beta$-tilt-stable if, for every $h\in\mathbb R^n$, the covariance operator of the tilted measure…

Vector-valued Borell inequality. If $0<\rho<1$ and

Let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^d$, and let $\mu$ be a log-Lipschitz perturbation of $\gamma$. Let $T=\nabla\varphi$ be the Brenier map transporting…

Quantum Bohnenblust–Hille conjecture. Fix $d\ge 1$. There exists $C_d>0$, depending only on $d$, such that for all $n\ge 1$ and every degree-at-most-$d$ operator…

Let $L_{\gamma,d}$ be the optimal constant in the Lieb–Thirring inequality, let $L_{\gamma,d}^{\mathrm{cl}}$ be the semiclassical constant, and let $L_{\gamma,d}^{\mathrm{one}}$ be…

Lieb–Solovej conjecture. If $p>2$, then