10 problems

Let $P$ be a simple $d$-polytope with $2d$ facets, and let $u$ and $v$ be two complementary vertices, meaning vertices not lying in a common facet. The $d$-step conjecture. There i…

Let $P$ be an integral polytope. A polytope is called a defect polytope when its associated projective toric variety has degenerate dual variety, equivalently when the invariant…

Let $P$ be an integral polytope, and let … where the sum runs over all nonempty faces $F$ of $P$. Nonnegativity conjecture. The invariant $c(P)$ is nonnegative: … Numerical experim…

Siegel's Conjecture. If $H$ satisfies the Hirsch Conjecture and $C$ is a cube, then $H\cap C$ satisfies the Hirsch Conjecture.

Let $clP_k$ denote the feasible region of density vectors of consecutive permutation patterns of size at most $k$, and let $\mathcal{L}_k$ be the set of Lyndon permutations of size…

McMullen–Shephard conjecture. Every projectively unique $4$-polytope is accounted for in Shephard's list of $11$ combinatorial types of projectively unique $4$-polytopes.

Shephard's conjecture. For every $d\ge 0$, every combinatorial type of $d$-dimensional polytope can be realized using subpolytopes of $d$-dimensional stacked polytopes.

Let $P$ be a $d$-dimensional polytope with $n$ facets, and let its diameter be the maximum length of a shortest edge path between two vertices of $P$. Hähnle's sharpened conjecture…

Let $P$ be a $d$-dimensional polytope with $n$ facets, and let its diameter be the maximum length of a shortest edge path between two vertices of $P$. Hähnle's conjecture. The diam…

For a polytope $P$ of dimension $d$ defined by $n$ inequalities and a linear objective function $c$, let $\lambda_c(P)$ be the total curvature of its central path. Let…