13 problems

The above-threshold product-structure conjecture. For every $\alpha \geq s(n)$, the class of intersection graphs of $\alpha$-free homothetic regular $n$-gons has product structure.

Let a Sharygin triangle be a non-isosceles triangle whose bisectral triangle, formed by the intersections of its internal angle bisectors with the opposite sides, is isosceles. Sup…

Let $N>6$ and consider a configuration of grasshoppers at the vertices of a regular $N$-gon, with legal moves given by the grasshopper-jumping game described in the paper. Polynomi…

Let an $n$-gon be a regular polygon considered in the space of plane group packing configurations, and let a $p2$ packing mean a packing with plane group symmetry $p2$. Pessimal he…

Isosceles-triangle boundary conjecture. The family filling the left boundary of the diagram $f(\mathcal K^2,P)$ is given by the triangles $T_\lambda$ for $\lambda\in[0,1/2]$.

Pena et al.'s conjecture. The quantization coefficient for the uniform distribution on the boundary of a regular $m$-sided polygon inscribed in a circle is an increasing function o…

Let $S^1$ be the circle of unit radius in $\mathbb{R}^2$ centered at the origin, and let $P_n$ be an inscribed regular polygon with $n$ sides; equip both with the Euclidean metric.…

Let $P_n$ be a regular polygon with $n$ sides, let $s_{2l+1}$ denote the side length function for inscribed $(2l+1)$-pointed stars, and call a point of $P_n$ a midpoint crossing or…

Let $S_n$ be the slack matrix of the regular $n$-gon, and let $\operatorname{rank}_{\operatorname{psd}}(S_n)$ denote its positive semidefinite rank. Logarithmic psd-rank conjecture…

For each $N\geq 3$, let $P_N$ be a regular $N$-gon, let $\operatorname{Area}(P_N)$ denote its area, and let $\lambda(P_N)$ be its first Dirichlet Laplacian eigenvalue. Ratio monoto…

For each $N\geq 3$, let $P_N$ be a regular $N$-gon, let $\operatorname{Area}(P_N)$ denote its area, and let $\lambda(P_N)$ be its first Dirichlet Laplacian eigenvalue. Regular-poly…

Regular polygon conjecture. Then $A_1,A_2,\ldots,A_n$ are the vertices of a regular polygon inscribed in a circle concentric to $\Gamma$.

Let $A_1,\ldots,A_n$ be $n$ different points in the plane, and let $\Gamma$ be a circle such that, for every $k\in\\{1,\ldots,n-1\\}$, the sum $\sum_{i=1}^n PA_i^{2k}$ is independe…