26 problems

Let $P$ be a polytope described by $n$ inequalities in dimension $d$, and consider its vertex-edge graph. Hirsch conjecture. The diameter of the vertex-edge graph of $P$ is at most…

Let $d$ be the dimension of the classical associahedron. Sleator–Tarjan–Thurston conjecture. Its diameter is equal to $2d-4$ as soon as $d>9$. This conjecture concerns the exact di…

A polytope is neighborly if every set of at most $\lfloor d/2 \rfloor$ vertices forms a face. For any pair $(n,d)$, a cyclic polytope is a $d$-dimensional polytope with $n$ vertice…

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…

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…

Hetyei's conjecture. There are cubical $4$-polytopes whose $2$-skeleton is not edge-orientable.

Let $\operatorname{Cosmo}_{n-1}$ denote the polytope defined by Arkani-Hamed, Figueiredo, and Vazão in the kinematic-space construction described in the source. Its combinatorial o…

A Matryoshka is obtained from a subdivision of a polygon by sequentially wrapping groups of polygons into larger polygons. The cosmohedron is the polytope underlying the cosmologic…

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…

A convex polytope is a bounded polyhedron; its diameter is the maximum graph distance between two vertices, where the graph has the vertices of the polytope and its edges. Let $P$…

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

Let $\textit{PPerm}(m,n)$ denote the partial permutation polytope. Volume conjecture. The normalized volume of $\textit{PPerm}(n,2)$ (equivalently, $\textit{PPerm}(2,n)$) is equal…

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…

A cubical polytope is a polytope in which every facet is a cube. Its dual graph has one vertex for each facet, with two vertices adjacent when the corresponding facets share a ridg…

An inscribed zonotope is a zonotope whose vertices lie on a common sphere. A zonotope is combinatorially equivalent to another polytope if their face lattices are isomorphic. A…

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 $P$ be a polytope, and let its $f$-vector be the vector $(f_0,\ldots,f_{d-1})$, where $f_i$ is the number of $i$-dimensional proper faces of a $d$-polytope. A vector…

Let $d\geq 2k+1$, let $S$ be a triangulated $d$-sphere with face numbers $(f_0,\dots,f_d)$, and let $f_j$ denote its number of $j$-faces. Generalized Lower Bound Conjecture. For ev…

Centrally symmetric realization conjecture. There is a $3$-polytope $P$ combinatorially equivalent to $Q$ such that

Let $V\subset\mathbb R^3$ be an extremal set, and let $\mathcal B(V)$ be its Ball polytope. Call $V$ strongly critical if it has no proper subset which is extremal. Strong critical…

Let $P$ be a $3$-polytope, and let $V\subset\mathbb R^3$ be an extremal configuration; in fact, the source restricts to critical configurations. Write $\mathcal F(P)$ for the face…

Let $V\subset\mathbb R^3$ be an extremal configuration. Call $V$ strongly critical if no proper subset of $V$ is extremal. Let $\mathcal B(V)$ denote its Ball polytope, and conside…