22 problems

Radial symmetry conjecture. The extremal functions in $(\mathrm{cknrs})$ are radially symmetric if

Widom's conjecture. The corresponding Szegő–Widom asymptotics holds for such sets, with a single arc behaving like an interval.

Let $\Delta(\varepsilon)$ be the function studied in the paper, with $0\leq\varepsilon\leq1$. Exact Delta conjecture. … for all $0\leq\varepsilon\leq1$. Determining…

For $k\geq1$, let $\Delta_k(\varepsilon)$ denote the function studied in the paper, with $0\leq\varepsilon\leq1$. Delta three and Delta four polygonal conjecture. The graph of…

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…

Extremal-profile conjecture. For general $p\neq 2$, up to translation and scaling, an extremal function is conjectured to have the form

Alvino–Ferone–Trombetti's conjecture. Every extremal function $u$ for inequality (1.1) is of the form

Brasco's classification conjecture. All minimizers for $S_{s,p}$ are of the form

For the classical Strichartz inequality corresponding to the case $\alpha=2$, let an extremal be a nonzero function attaining equality in the inequality. Gaussian extremals conject…

Let $d4a4^s(d6a9)$ denote the fractional Sobolev space on the domain $d6a9$, and let $d700_s(q)$ and $d4c0_s(q)$ be the two extremal functions defined in the source for $q>2$. Naza…

Fractional Hardy–Sobolev optimizer conjecture. Up to constant multiples, rescaling, and, when $\alpha=0$, translations, every Aubin–Talenti function is

McLeod–Peletier conjecture. There should exist a function $g$ satisfying these conditions such that $C_{g,4\pi}(\Omega)$ does not admit any extremal function. The conjecture is pos…

Let $\mathcal{A}_+(28)$ denote the class of radial Schwartz functions on $\mathbb{R}^{28}$ satisfying the positive Fourier-eigenfunction uncertainty conditions. Dimension-28 root-l…

Two-slope extremality conjecture. If the function has no uncovered components, then it is extreme.

Let $r,k\in\mathbb{Z}_+$ with $r>k$, and let $1\leq p,q\leq\infty$. Define … where the minimum is taken over … An extremal is a function attaining this minimum. Symmetry conjecture…

Extremal-function approximation conjecture.

Let $K$ be a convex body in $\mathbb{R}^{N}$, and define $\eta(K)$ as the infimum of $\int_{\mathbb{R}^{N}}F(\mathbf{x})\,d\mathbf{x}$ over nonzero continuous functions $F$ such th…

Let $K\subset\mathbb{R}^{N}$ be a convex body, and let $\eta(K)$ be the infimum of $\int_{\mathbb{R}^{N}}F(\mathbf{x})\,d\mathbf{x}$ over nonzero continuous functions $F$ satisfyin…

Let $t$ be a nonnegative integer, and let $\mathit{fw}(u)$ denote the formation width of a sequence $u$. Formation width conjecture. … The paper states that this would imply nearly…

Conjecture on $K_1(n)$. For all integers $n\ge3$,

Let $L^2(\mathbb{R})\cap\mathcal{C}(\mathbb{R})$ be the class of square-integrable continuous functions, and let an extremal mean an extremal of the cone of positive positive defin…

Let $\Omega_d^+$ denote the cone of positive positive definite functions on $\mathbb{R}^d$, and call $f\in\Omega_d^+$ an extremal when the interval it generates satisfies … A Gauss…