17 problems

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…

Regular polygon conjecture.

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 $P$ 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…

Let $n\geq 6$ be even, and let $A_n^*$ denote the maximal area among all small $n$-gons. An optimal $n$-gon is a small $n$-gon whose area is $A_n^*$. Its diameter graph is the grap…

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

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…

Let $n\geq 3$. An $n$-gon is a polygon with $n$ sides, and let $\lambda_1$ denote its first Dirichlet eigenvalue. Among all $n$-gons with fixed area, the regular $n$-gon is the pol…

Homogeneous boolean rank conjecture. For any $n$,

Ehrhart-function conjecture. There exists an equidecomposability relation $\mathcal{F}:(S,\mathcal{T})\to(S',\mathcal{T}')$ if and only if

Let Ehrhart-equivalent rational polytopes be rational polytopes whose Ehrhart functions agree. Discrete equidecomposability is allowed to use an arbitrary number of simplices, and…

Amongst all convex $n$-gons of fixed area, let the determinant refer to the determinant of the associated Laplacian. Regular polygon conjecture. Amongst all convex $n$-gons of fixe…

Let $P\subset\mathbb{P}^2$ be an axis-aligned $2n$-gon. Let $T$ denote the pentagram map, and let … be its point of collapse. Let $\mathscr C(P)$ denote the center of mass of $P$.…

Let $P(n)$ denote a polygon with $n$ sides and let $R(n)$ denote a regular polygon with $n$ sides. Write $A$ for area, $R$ for inradius, and $\lambda_1<\lambda_2$ for the first two…