146 problems

Voronoi's conjecture. Every parallelohedron is an affine transformation of a Dirichlet–Voronoi cell of some lattice.

Let $C$ denote the $n$-dimensional unit cube, let $k$ be a positive integer, and let $\Lambda$ be an $n$-dimensional lattice. A $k$-fold lattice tiling $C+\Lambda$ is a $k$-fold tr…

Let $P$ be a reflexive polytope of dimension $d$, and write $\lvert V(P)\rvert$ for its number of vertices. Nill's conjecture. The number of vertices satisfies … This is a proposed…

Let $\Diamond^n=\operatorname{conv}\{\pm e_1,\dots,\pm e_n\}\subset\mathbb R^n$ be the standard cross-polytope, let $E$ be a $k$-dimensional subspace, and let $P_E$ denote orthogon…

Let $e_3$ be the third standard basis vector, let $\mathbb{S}^2$ be the unit sphere, and consider the class $\mathcal{P}_{3,K}\cap\mathcal{I}_3$ of polytopes with $K$ vertices. For…

Let $\Omega$ be a finite family of convex polytopes in $\mathbb{R}^n$. For each unit vector $g\in S_{n-1}$, define … Set … and, starting from $\Omega_0$, define…

A perfect Delaunay polytope is a perfect Delaunay polytope in dimension $d$, and let $ED_d$ denote Dutour's polytope. Dutour's conjecture. The polytope $ED_d$ has the largest numbe…

A Petrie scheme of an abstract polytope is the shortest cycle or bi-infinite flag sequence generated by a product of the flag-adjacency involutions in a permutation of all ranks; i…

Talata's conjecture. The number of vertices of an edge-antipodal $3$-polytope is bounded above by a constant.

Dihedral-angle rigidity conjecture. A convex polytope is determined up to congruence by its dihedral angles.

Let $P\subset\mathbb{R}^d$ be a centrally symmetric polytope, meaning that $x\in P$ implies $-x\in P$. It is 2-neighborly if every pair of non-antipodal vertices is the vertex set…

Dihedral-angle uniqueness conjecture. A polytope is determined up to congruence by its dihedral angles.

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…

Classification conjecture. If $\Gamma$ is isometrically embeddable in a half-cube, then $\Gamma$ occurs in Table 3, Table 4, or in one of the infinite series discussed in this sect…

Uniqueness conjecture. There are no other extreme Delaunay polytopes in dimension $7$.

Bressler–Lunts hard Lefschetz conjecture. For every $i<d/2$, this map is a bijection. This conjecture proposes the hard Lefschetz property for intersection cohomology of arbitrary…

Let ${\cal P}$ be a convex polytope, and let $h({\cal P})=(h_0,h_1,\ldots,h_d)$ be its toric $h$-vector. Define the associated $g$-vector by the successive differences of the first…

Consider a two-dimensional slip system with convex polytope … where $\det(S^\alpha)=0$ and $0\in\operatorname{int}(\mathcal{U})$. Let $f_0,f_1,f_2$ be the numbers of vertices, edge…

Let $P$ be a simple $n$-dimensional convex polytope, and let $f_{n-k}$ denote its number of $(n-k)$-dimensional faces. Write … where the $\beta_{\ell}$ are positive integers, and s…

Mirković–Vilonen's inductive construction conjecture. There is an inductive construction of the polytopes $P_\phi$ for any semisimple group.

Zaks's finiteness conjecture. The largest neighborly family of congruent $3$-polytopes is finite.

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 $P(\Phi)$.

Dimension-square mixing conjecture. The mixing time could be