28 problems

Let $n\geq6$ be an even integer, and let $P$ be an optimal small $n$-gon. Its skeleton is the graph consisting of the vertices of $P$ together with all line segments joining pairs…

Graham's conjecture. For every even $n\ge6$, the diameter graph of every optimal $n$-gon consists of an $(n-1)$-cycle together with one additional edge emanating from one of the cy…

Let $\Pi_n$ be an $n$-gon, with perimeter $L_n$, area $A_n$, and side lengths $a_1,\ldots,a_n$. The source considers the hypothesis that $\Pi_n$ is cyclic and the associated quanti…

Zhang's pseudo-perimeter conjecture. For any cyclic $n$-gon, the inequalities stated in the source hold for the quantities introduced above. The source excerpt supplied here does n…

Let $\Pi_n$ be an $n$-gon with side lengths $a_1,\ldots,a_n$, perimeter $L_n$, and area $A_n$. Define … and set $\varphi_n=A_n/P_n$. Levy's ratio bounds conjecture. For any $n$-gon…

Let $\Pi_n$ be an $n$-gon with side lengths $a_1,\ldots,a_n$, perimeter $L_n$, and area $A_n$. Define its pseudo-area by … For the regular $n$-gon, let … Define the ratio…

Let $P$ be an ideal hyperbolic $k$-gon, with its sides labelled cyclically, and let $\mathbf{a}$ be a billiard sequence. Let $\gamma_{\mathbf{a},P}$ be the collection of all closed…

Let $P\subset H^2$ be an ordinary reduced $n$-gon of minimal width $w$. Let $\widetilde{P}$ denote the regular ordinary reduced $n$-gon of width $w$, and write…

Let $(\mathcal{Q}^{(n)},F|_{Y^{(n)}})$ be the piecewise isometry considered in the paper, with atoms given by the partition $\mathcal{Q}^{(n)}$. Finite convex refinement conjecture…

Consider the heat content functional for an $N$-gon under a fixed-area constraint, with $t\in\mathbb R$, and let the regular $N$-gon be the symmetric critical configuration. Local…

Fix $Q=12$ and consider the Hessian of … where $\ell_Q$ is chosen so that the regular $N$-gon is a critical point. Restrict to eigenvectors orthogonal to the gradient of the area;…

Let $J_h$ be the polygonal Riesz functional on the class $\mathcal P_N$ of $N$-gons, and call a polygon critical for $J_h$ when its first variation under the elementary side rotati…

Decagon perimeter conjecture. One has

Nikitenko's conjecture. Any extremal polygon for odd $n$ is a regular $n$-gon.

Strut-side perimeter conjecture. The perimeter $L(P)$ satisfies

Generic polygon image conjecture. The map $\alpha: \mathsf{TS}_{k}\rightarrow\mathsf{TP}_{k+6}$ has image in $\mathsf{TP}^{\mathsf{gen}}_{k+6}$ and is a homeomorphism onto its imag…

Let $\mathcal{J}$ be the functional on the class $\mathcal{P}_n$ of simple polygons used in the paper. A polygon is a critical point of $\mathcal{J}$ when the limit in the paper's…

Let $\mathcal{J}$ be the functional on the class $\mathcal{P}_n$ of simple polygons used in the paper. For $n\in\{3,4\}$, the maximizers of $\mathcal{J}$ over $\mathcal{P}_n$ are r…

Let $N\geqslant 3$ be a fixed integer, and consider polygons with $N$ sides in the setting of Theorems 2 and 3, concerning the relevant Riesz-type inequalities and overdetermined p…

Let $Delta$ and $Delta'$ be planar polygons with corresponding vertices $xi,wj,uk,vl$ and $xi',wj',uk',vl'$, and let a subpolygon comparable to $Delta$ mean a subpolygon…

Let $Delta$ and $Delta'$ be planar polygons with three strictly convex vertices $x1,x2,x3$ and $x1',x2',x3'$, respectively, and let corresponding vertices be defined as in the pr…

Polya's conjecture. The $n$-gon of area $1$ with maximal torsional rigidity is the regular $n$-gon of area $1$.

Average-length conjecture. For every ideal hyperbolic polygon and every family of cyclically related closed billiard trajectories, the average length function is uniquely minimized…

Let a pentagon be iteratively transformed by the $cP$-evolute operation, where the $cP$-evolute is the polygon whose sides are the perpendicular bisectors of the original polygon's…

Gutkin's conjecture. The listed condition is necessary for a polygonal billiard table to be secure.