251 problems

Grünbaum's conjecture. The function $\theta_k(d+s,d)$ gives the minimum number of $k$-faces of a $d$-polytope with $d+s$ vertices.

McMullen–Shephard's conjecture. For every $n\geq 3$, one has

Let $V=\{x\in\mathbb R^{d+1}:\sum_{i=0}^d x_i=0\}$, and for vectors $v_i$ spanning $\mathbb R^d$ let … A convex set is a universal cover if every subset of $\mathbb R^d$ of diamete…

Let $X_G$ be the chromatic symmetric function of a graph $G$, and let $X_G(x_1,\ldots,x_k)$ denote its specialization to $k$ variables. A polynomial is SNP (has a saturated Newton…

Jonsson's polytopality conjecture. For every $n\geq 2k+1$, the complex $\overline{\mathcal{A}\hskip-0pt\mathit{ss}}_{k}(n)$ is a polytopal sphere: there is a simplicial polytope of…

Let $k\geq 1$ be an integer, and let $\mathcal{K}_k$ be the simplicial complex whose facets are maximal “$k$-crossing-free” sets of diagonals of a convex polygon, called multi-tria…

Let $P$ be a simple polytope, and let $P^*$ be its polar dual. A Perles piece is a connected simplicial complex such that every facet contains exactly one ridge in the boundary of…

Let $d\ge 2$ be fixed. A separator of a graph on $n$ vertices is a partition of its vertex set into sets $A,B,C$ with $cn\le |A|\le |B|\le (1-c)n$, for a fixed constant $c$ satisfy…

Let $\Delta\subset\mathbb{R}^d$ be a polytope, let $\Phi$ be its dual fan, and let $\overline{M}_\Phi$ be the quotient combinatorial intersection cohomology module defined from the…

Let $\Delta$ be a $d$-dimensional polytope, and let $Gh_k(\Delta)$ denote the components of Stanley's generalized $h$-vector. For a polytope that is integral, these numbers agree w…

Let $P$ be a primitive polytope in $\mathbb{E}^d$, where $d\geq 4$. Primitive-polytope vertex conjecture. The polytope $P$ has at most $2^d$ vertices, and it has fewer than $2^d$ v…

Let $\Delta^n$ be the $n$-simplex, and let $h_{\Delta^n}$ denote its support function, defined by … The depth of a neural network is the number of hidden layers required to compute…

A classical symmetric edge polytope is the symmetric edge polytope associated with a graph, and its $h^\ast$-polynomial is the Ehrhart $h^\ast$-polynomial of that polytope. Gamma-p…

Let $d\geqslant 3$ and $1\leqslant\ell\leqslant d-1$. Let $P$ be a $d$-polytope with $2d+\ell$ vertices, and let $T(r)$ denote the $r$-simplex. Pineda-Villavicencio's conjecture. 1…

Brenti–Welker's conjecture. The chain polynomial of the face lattice of every polytope is real-rooted.

Let $P$ be a regular polypol, meaning that its boundary contains no singular points of its ambient variety $Y$ except for the vertices and that…

For each sequence $s$, let the $s$-permutahedron be the generalized permutahedral object defined in the source. A polytopal subdivision is a subdivision whose cells are polytopes.…

Möbius uniqueness conjecture. Every edge-scribable $(d+1)$-polytope is Möbius unique.

For any integer $k 1$, let a $(k,\nu)$-tree be a maximal subset of $A_\nu$ without $k+1$ pairwise $\nu$-incompatible elements, and let the $(k,\nu)$-Tamari complex be the simplicia…

Consider the poset of polytopal subdivisions in a fixed class induced by a projection of polytopes, including the coherent subdivisions whose poset is the face lattice of a fiber p…

Erickson's conjecture. There are no $4$-polytopes or $3$-spheres on $n$ vertices with $\Omega(n^2)$ non-simplicial facets.

Isoperimetric optimality conjecture. The 120-cell has the least volume among all 4-polytopes of unit in-radius having 120 cells.

Enumeration conjecture for ${\cal A}_n^j$. The number of elements of ${\cal A}_n^j$ is

Let $K$ be a convex body of constant width in ${\mathbb{R}}^4$, and let $C$ be a regular cross polytope, also called a generalized octahedron. Makeev's conjecture. Every constant-w…

Let $K$ be a convex body of constant width in ${\mathbb{R}}^n$, and let $D_n$ denote the dual of the difference body of a regular simplex. Makeev's conjecture. Every constant-width…