90 problems

Let $G$ be a finite connected graph, let $d$ be its diameter, let $W(G)$ denote its Wiener index, and let $C_{2d+1}$ denote the cycle on $2d+1$ vertices. DeLaViña–Waller conjecture…

Frank–Loss conjecture. The infimum $S_3$ is achieved by $\xi$.

There are $N$ coupon types, with draws made independently according to a probability vector $\mathbf{p}=(p_1,\ldots,p_N)$ in the open simplex … and let $T_N$ be the first time ever…

Equal-weights conjecture. For every $x$, the supremum in the definition of $F(x)$ is attained by a Rademacher sum having all of its weights equal.

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

Let $P$ be a $k$-term partition, and let $\accentset{\text{\large\bfseries.}}{K}(\cdot)$ denote the semi-regular continuant associated with an arrangement of the parts of $P$, whos…

Let $\Sigma_0$ be the class of univalent meromorphic functions $F$ in the exterior of the unit disk, normalized by $F(z)=z+O(z^{-1})$ as $z\to\infty$. Let $K=F(\mathbb{B}^c)^c$ con…

Let $C=B_2^k(0,w)\times\mathbb{R}^l$, where … Let $K$ be a centrally symmetric convex body with $\gamma(K)=\gamma_{k+l}(C)$ and with inradius $w(K)=w=w(C)$, where $\gamma_{k+l}$ is…

Let $\Delta(\varepsilon)$ denote the symmetric subset function studied in the source, with $0\leq\varepsilon\leq1$. Exact formula conjecture for $\Delta$. … for all…

Schinzel–Schmidt minimizer conjecture. Among all probability density functions supported on $[0,\frac12]$, this function minimizes $\|f\ast f\|_\infty$.

Even-cycle degree-power Turán conjecture. For every $p>1$,

Let $B_\infty^n$ denote the unit ball of $\ell_\infty^n$, and for $K\in\mathcal{K}_s^n$ let $U_K=U_{K,n+1}$ and $V_K=V_{K,n}$ be the normalized volumes of a random simplex and rand…

Let $\mathcal{K}^n$ be the family of convex bodies in $\mathbb{R}^n$, let $\mathcal{K}_s^n$ be the family of symmetric convex bodies, and let $U_{K,N}$ and $V_{K,N}$ denote the nor…

A probability density function (pdf) supported on $[0,\frac12]$ is nonnegative and integrates to $1$. For such a function, write $f\ast f$ for its autoconvolution and…

Lassak's conjecture. Every planar reduced body $R$ satisfies

Center–median conjecture. The center and median of $T_0$ have distance zero:

Finite variance extremality. The equal-probability vector $\nu$ minimizes

Let $n\geq 3$ and consider extremizers of the $p$-Stam inequality among normalized monic real-rooted polynomial pairs $(f,g)$ of degree $n$; write $d_{PQ}$ for their pair mismatch…

Dubickas' conjecture. Every elementary symmetric function of these quantities attains its maximum when the $\omega_j$ are the vertices of a regular $n$-gon inscribed in…

Extremal-constant problem. In the previous problem, $c=q_\infty$.

Let $\mathbb{T}$ be the unit circle, and for polynomials $P$ of degree at most $d$ let $\mathscr{C}_{d,p}$ be the smallest constant $C$ such that … where the $L^p(\mathbb{T})$ norm…

The linear upper-bound conjecture. If $d\geq 1$ and $p\geq 2$, then

Let $\Omega=\mathbb{D}$ be the unit disk, let $\Gamma=\partial\Omega=\mathbb{S}^1$, and let $\mathcal{I}_\alpha^n\subset\mathbb{S}^1$ be the union of $n$ evenly spaced intervals of…

Steinerberger's conjecture. The characteristic function of the interval is a global extremizer of this uncertainty principle. Steinerberger proved that the characteristic function…

Consider configurations of zeros used to form monic degree-$n$ polynomials and their lemniscates $\Lambda_p=\{z\in\mathbb C:|p(z)|<1\}$. The numerical minimizers are configurations…