5 problems

Bounded convex decomposition conjecture. There exists a map

Let $\rho$ be a parameter, let $\Lambda_\rho$ be the associated set, and let $R_\rho(\Lambda_\rho)$ denote its roundness. For each $k\in\mathbb{N}\setminus\{0,1\}$, consider the pa…

Diagonal monotonicity conjecture. If $0\leq\alpha\leq\tilde\alpha$, then

Let $f\colon\mathbb{T}^2\to\mathbb{T}^2$ be a homeomorphism homotopic to the identity and let $\tilde f\colon\mathbb{R}^2\to\mathbb{R}^2$ be a lift of $f$. Suppose that the rotatio…

Let $f\in{\mathrm{Homeo}}_0(\mathbb{T}^2)$ have a rotation set $\rho(F)$ with non-empty interior, and let $C\subseteq{\mathrm{int}}(\rho(F))$ be a continuum. A realization conjectu…