12 problems

Cubical-polytope reconstruction conjecture. Every cubical polytope can be reconstructed from its dual graph.

Centrally symmetric cyclic upper bound conjecture. If $d$ is even, then the boundary complex $\partial C_d^{\Delta}$ of the $d$-dimensional crosspolytope on $n=2d$ vertices is the…

A $(d,n,m)$-maxout polytope is a polytope specified by the parameters $d$, $n$, and $m$, and $f_0(P)$ denotes the number of vertices of a polytope $P$. Maxout polytopes vertex-numb…

Spherical orbit-space conjecture. The space $N(P,\lambda)$ is homeomorphic to $S^n$ if and only if

Let $\mathcal{P}(m,n)$ denote the partial permutohedron. Volume conjecture. The normalized volume of $\mathcal{P}(m,2)$ is equal to $3^m-m$. This conjecture gives a closed formula…

Let $\mathcal{P}(m,n)$ be the partial permutohedron and let $\mathcal{B}_m$ be the Boolean lattice of subsets of $[m]$. For a chain in $\mathcal{B}_m$, the difference between its l…

Let $P$ be a $d$-dimensional polytope with $n$ facets, and let ${\bf c}$ be a generic linear functional. Orient the graph of $P$ according to increasing values of ${\bf c}$, and de…

Let $P$ be a $d$-dimensional polytope, and let $u$ and $v$ be any two of its vertices. A path is nonrevisiting if, after leaving a facet, it does not later return to that facet. No…

Let $P$ be a simple polytope and let ${\bf c}$ be a generic cost vector such that $G(P, {\bf c})$ is the Hasse diagram of a lattice. A directed path in $G(P,{\bf c})$ is said to re…

Monotonicity conjecture. $d719$ is monotone if and only if $d719$ is CM-monotone.

Let $K$ be a cubical polytope of dimension $d\geq 3$, with $V$ vertices and $E$ edges. Jockusch's conjecture. … The source attributes this conjecture to Jockusch and discusses part…

Let $d$ and $k$ be integers with $d\geq 2k+1$, and let $X$ be a triangulated $d$-sphere with face-vector $(f_0,\dots,f_d)$. Generalized lower bound conjecture. For every $j$, the f…