3 problems

Optimal packing conjecture. The packing by Reuleaux triangles has the greatest density among packings of the plane by a curve of constant width.

Let $N\geq 2$, and let $p$ be a trigonometric polynomial of degree $N$ with coefficients $a_N,b_N$ satisfying $a_N\ne 0$ or $b_N\ne 0$. Let $f\in\mathbb{R}[x,y,z]$ be the defining…

Let $\mathcal{C}$ be a planar algebraic curve of constant width whose support function is a trigonometric polynomial of degree $3$. Apart from the circle, whose defining polynomial…