8 problems

Equal-density packing conjecture. Densities of the densest $p2$, $pg$, and $p2gg$ packings are equal for all, but centrally nonsymmetric $n$-gons with three-fold rotational symmetr…

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 $S^1$ 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…

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