14 problems

Let $P$ be a polytope. A path $P'$ in its graph is nonrevisiting if, for every facet of the polytope, the intersection of $P'$ with that facet is either empty or a path in the face…

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…

A net, or unfolding, of a 3-polytope is obtained by cutting it along edges so that the resulting connected surface can be flattened into the plane. A net is nonoverlapping if its p…

A polytope is decomposable if it is a Minkowski sum of dissimilar convex bodies; two polytopes are similar when one is obtained from the other by a dilation and a translation. The…

Let $P$ be a $d$-dimensional polytope with $f$ facets. Its circuit diameter is the maximum, over pairs of vertices, of the shortest number of steps in which one may move along a ci…

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.

Bounded Hirsch conjecture. The diameter of the facet-ridge graph of any $d$-polytope on $n$ vertices is $\le n-d$.

Hirsch's conjecture. El diámetro combinatorio del grafo de un politopo de dimensión $d$ definido por $n$ desigualdades no puede ser nunca mayor que $n-d$.

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…