4 problems

Face-structure conjectures for tropical polytopes. The following assertions hold: (1) $k$-faces of tropical polytopes are extreme sets; (2) the topological boundary of a $k$-face i…

Tropical halfspace characterization conjecture. A tropical polytope $P$ is pure and full dimensional if and only if it has a halfspace description whose apices are in general posit…

Tropical halfspace representation conjecture. If $P$ is pure, then the halfspaces from a generic lift $\overline{P}$ of $P$ map to tropical halfspaces whose intersection is $P$ its…

Let $\mathbb{T}\mathbb{N}$ denote the tropical natural numbers, let $M \in \mathbb{T}\mathbb{N}^{d \times m}$, and let $P=\operatorname{tconv}(M)$ be the corresponding tropical lat…