33 problems

Let $m$ be a positive integer and consider all $m$-gons of a fixed area. The affine-orbit part of the Pólya–Szegő polygonal conjecture. The regular $m$-gon minimizes the first Diri…

For each integer $N\geq 3$, let $P_N$ be a regular $N$-gon whose area is fixed, and let $\lambda_1(P_N)$ denote its first Dirichlet eigenvalue. Antunes–Freitas conjecture. The sequ…

Let a small $n$-gon be a convex polygon in the plane with $n\geq 3$ vertices and diameter at most one. An optimal small $n$-gon is one attaining the maximum possible perimeter amon…

Let $P$ be a strictly convex polygon in $\mathbb{RP}^d$. Suppose that $P$ intersects a hyperplane with multiplicity $k$, where $k\geq d+2$. Call a vertex configuration a flattening…

Let $P$ be an embedded closed polygon in $S^2$ that bisects the area, and call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. Discre…

Let $P$ be an embedded non-contractible closed polygon in $\mathbb{RP}^2$, and call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. D…

Let $P(m,n)$ be the product of an $m$-gon and an $n$-gon, and let $\lVert P(m,n)\rVert$ denote its polytopal Gromov norm. Polytopal Gromov norm conjecture. $\lVert P(m,n)\rVert$ sh…

Let $P(m,n)$ be the product of an $m$-gon and an $n$-gon, and let $\lVert P(m,n)\rVert$ denote its polytopal Gromov norm. Polytopal Gromov norm conjecture. $\lVert P(m,n)\rVert$ sh…

Let $\mathscr{P}$ be a simple polygon. A witness set for $\mathscr{P}$ is a set of points in $\mathscr{P}$ such that every point of $\mathscr{P}$ is visible from at most one point…

Let $P$ be a polygon in $\mathbb{R}^2$, and let a side mean an edge of $P$. For a positive integer $p$, let $\omega_p(P)$ denote the $p$-width of $P$. The single-side $p$-width con…

Let $P_n$ be a small $n$-gon, meaning a convex polygon in the plane with $n\geq 3$ vertices and diameter at most one. Its perimeter is denoted by $p(P_n)$. An optimal small polygon…

Let a small $n$-gon be a convex polygon in the plane with $n\geq 3$ vertices and diameter at most one. A diameter chord is a side or diagonal of length one. For even $n$, the conje…

Diameter minimality conjecture. If $V$ is non-regular, then

Let $\Omega$ be a planar region, let $\rho_N(\Omega)$ be the squared distance from $\bar z$ to the polynomials of degree at most $N$ in $L^2(\Omega,dA)$, and call the corresponding…

Let an equilateral triangle and a square have the same area, and consider dissections into finitely many pieces that may be rearranged, with pieces allowed to be turned over. Three…

Let $f(n,m)$ denote the maximum number of overlaps between an $n$-gon and an $m$-gon. Intersecting polygons conjecture. The answer to the problem of determining $f(n,m)$ is … where…

Let $P$ be a $(k+3)$-gon in $\mathbb{RP}^k$, where $k\geq 2$, and let $J=(1,\ldots,1)$. For even $k$, let $I=(1,\ldots,1,2,1,\ldots,1)$ with the same number of $1$'s before and aft…

Pedal-polygon equivalence conjecture. For integer $n\ge 5$, if $G(n)$ is the set of $n$-gons, then

Kuperberg's conjecture.

Let $n\geq 3$, and let $\Omega_1$ and $\Omega_2$ be convex $n$-gons. A primitive closed geodesic is a closed geodesic that is not a nontrivial iterate of a shorter closed geodesic;…

Let $n\geq 3$, let $\Omega$ be a convex $n$-gon, and let $P$ be a regular $n$-gon. Write $\lambda_i(\Omega)$ for the $i$th eigenvalue of the Dirichlet Laplacian on $\Omega$, and le…

Distance conjecture. The upper estimates obtained for these distances are exact:

Let $K$ be a convex pentagon in the plane, and let $K_1$ be the pentagon bounded by the diagonals of $K$. Affine regular pentagon conjecture. The maximum of the ratio between the a…

A polygon is an $n$-gon of fixed area. Pólya's conjecture. Among $n$-gons of fixed area, the regular one minimizes the first Dirichlet eigenvalue. The conjecture is used as a hypot…

Homogeneous boolean rank conjecture. For any $n$,