105 problems

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…

Chang--Yang conjecture. For every $\alpha\geq\frac12$ and every $u$ satisfying

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

Log-Sobolev KLS conjecture.

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

Endpoint sharp-constant conjecture. In the endpoint case $p=1$,

Higher-order Rellich inequality conjecture. One has

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

Talagrand's conjecture. For any $s>0$, there exists a constant $c_s$ independent of the dimension $n$ such that, for $t>1$,

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 $\gamma$ 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…

Sharp four-dimensional PES threshold. The free energy $\mathcal F_{\mathrm{PES}}$ should be bounded below on $\mathcal A_M(\Omega)$ for $M<M_*$, should lose compactness at $M=M_*$,…

Step-size scaling conjecture. The $O(h^4)$ scaling of the spectral gap is merely an artifact of the isoperimetric approach.

Let $K\text{ be a convex body in }\mathbb{R}^n$ 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…

Optimal Prékopa–Leindler stability conjecture. There exists a constant $C(n)>0$, depending only on $n$, such that the corresponding stability estimate holds.

Given probability measures $\eta,\gamma\in\Pi(\mathrm{Leb},\mathrm{Leb})$, let $(c_t,v_t)_{0\leq t\leq1}$ be a minimizer of the multiphase incompressible optimal transport problem,…

de Figueiredo–do Ó–Ruf conjecture. The supremum $S_{\Omega}\left(\lambda,\frac{N}{N-1}\right)$ should not be attained when $\lambda\geq\alpha_N$. They established attainment for ev…

The functional B-conjecture. The function $\upalpha:\mathbb R\to\mathbb R_+$ is log-concave.

Martin–O'Bryant's conjecture. One has

Symmetry-breaking conjecture. The only manner in which the symmetry of a global extremal can break is when the symmetric extremal function is linearly unstable; consequently, it ca…

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 $\mu_1$ be the double-sided exponential measure on $\mathbb{R}$ with density $e^{-\lvert x\rvert}/2$. For an absolutely continuous function $f$, let $f_k$ denote its polynomial…

Let $s\in(-\frac{1}{n},0)$ and let $g:\mathbb{R}^{n}\to\mathbb{R}_{+}$ be an even $s$-concave function such that $0<\int g<+\infty$. Let $\mathcal{L}_{s}$ denote the $s$-polarity o…