59 problems

Let $P$ be a $d$-polytope with vertex set $V$, and let $d'\le d-1$. Suppose that for every integer $k$ with $0\le k\le d'$, condition (dfcon) is satisfied. Write $i(P,m)$ for the E…

Let $\mathcal{I}_7$ denote the class of seven-vertex polytopes inscribed in the unit sphere $\mathbb{S}^2$, and let $S(P)$ be the surface area of $P$. Let $e_3$ be the third standa…

Let $P$ be a convex $d$-polytope with $f$-vector $(f_0,f_1,\ldots,f_{d-1})$, where $f_k$ is the number of $k$-dimensional faces. The face-number lower-bound conjecture. For every…

Let $P$ be a convex $d$-polytope with $f$-vector $(f_0,f_1,\ldots,f_{d-1})$, where $f_k$ denotes the number of $k$-dimensional faces. Bjrner's quarter-monotonicity conjecture. The…

Let $P$ be a convex $d$-polytope with $f$-vector … where $f_k$ is the number of $k$-dimensional faces of $P$. The unimodality conjecture. For each $d$-polytope $P$ there is an inte…

Cluster complex face-lattice conjecture. This poset is the face lattice of a simple $n$-dimensional convex polytope $P(\Phi)$.

Cluster fan polytopality conjecture. The simplicial fan generated by the clusters is the normal fan of a simple $n$-dimensional convex polytope $P(\Phi)$.

Algorithmic verification criterion. The following conditions are equivalent: the configuration $\lambda K+\mathbb{Z}^3$ is a lattice covering realizing the specified combinatorial…

Symmetric minimal covering body classification conjecture. There exists a unimodular transformation $T$ such that

The elements of a configuration are points in the $(m-1)$-dimensional probability simplex in $\mathbb{R}^m$. A configuration is extremal if it arises as the configuration associate…

Nill's conjecture.

Let a polytope be resistance-positive when its resistance curvature is positive at every vertex. Finiteness conjecture for resistance-positive simple 3-polytopes. There are finitel…

Let $P$ be a polytope, and call it Forman–Ricci-positive when its Forman–Ricci curvature satisfies $\kappa_F(e)>0$ for every edge $e$. Finiteness conjecture for Forman–Ricci-positi…

Let $P$ be a centrally symmetric $d$-dimensional polytope. A flag of $P$ is a chain of faces of $P$, one in each dimension. Kalai's full flag conjecture. $P$ has at least…

Let $\mathcal P$ be the combinatorial type of a convex polytope of dimension $d\geq 3$. Generic first-order rigidity conjecture. If $\mathcal P$ is the combinatorial type of a conv…

Let $n$ be a positive integer, let $\mathcal{P}^n$ denote the family of polytopes in $\mathbb{R}^n$, and let $\mathcal{M}(\mathbb{R}^n)$ denote the space of real-valued functions o…

Dolorfino et al.'s bounded-hypercube conjecture. If $\{M(P_i)\}_i$ is bounded, then

Let $P$ be a convex polytope with $n$ vertices and full-dimensional span in $\mathbb R^d$. Let $(G^*,\hat{\bf p})$ be the tensegrity framework on $n+1$ vertices obtained by putting…

Adjoint non-degeneracy conjecture. For a generic $n$-gon $Q=B^T\cdot P_b$, the adjoint $\alpha_Q$ is non-degenerate, with Newton polytope equal to the Newton polytope of

Wachspress ML-degree conjecture. For a generic $n$-gon $Q$, the maximum likelihood degree is

Irreducibility conjecture. For generic matrices $A \in \mathbb{R}^{d \times n}_{\geq 0}$ and each cell $C \in \mathcal{C}_A$, there exists a dense open subset…

Let $P$ be a simplicial $d$-polytope with $2d$ vertices. Two facets are estranged if they are disjoint. Von Stengel's conjecture. The maximum number of pairs of estranged facets of…

Let $Z$ be a zonotope, and let $\operatorname{rk}(Z)$ denote its rank, namely the number of summands in a representation of $Z$ as the Minkowski sum of a nondegenerate collection o…

An Archimedean solid is a three-dimensional convex polytope whose faces are regular polygons and whose symmetry group acts transitively on its vertices. Archimedean solid perfectio…

Let $P\subset\mathbb{R}^d$ be a flexible polytope, meaning that it admits a continuous deformation preserving its combinatorial type and edge lengths that is not a continuous isome…