33 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.

Let $\Delta$ be a simplicial $(d-1)$-sphere. The $g$-numbers are certain alternating weighted sums of the $f$-numbers. McMullen–Walkup conjecture. For any $k\leq d/2$, one has … Th…

Klee's conjecture. The upper bound conjecture holds for all Eulerian complexes.

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…

Let $P$ be a balanced simplicial $d$-polytope, meaning that its underlying graph is $d$-colorable. For $-1\leq i\leq d-1$, let $f_i(P)$ be the number of $i$-dimensional faces, with…

Let $d=2k$ be a fixed even dimension, and let $\mathcal{B}_{2k}(X)$ denote the centrally symmetric polytope constructed from a set $X$ of $n$ equally spaced points on…

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…

Let $k\geq 2$, $d\geq 2k$, and let $\Delta$ be a $(d-1)$-sphere with $g_k(\Delta)=1$. Cell-complex realization conjecture. There exists a $d$-dimensional cell complex $\mathcal{C}$…

Let $d$-polytopes have at least $d+3$ facets and between $3d$ and $4d-4$ vertices. For $d=4$, the condition $4d-4=3d$ specifies the endpoint with $12$ vertices. Four-dimensional fa…

Let $d\geqslant 5$, and consider $d$-polytopes with at least $d+3$ facets and between $3d$ and $4d-4$ vertices. A simple edge is an edge whose endpoints are both simple vertices. A…

Let $d\geqslant 2$ and $1\leqslant\ell\leqslant d-1$. Let $J(\ell+1,d)$ be the $d$-polytope obtained by truncating a simple vertex from the $(\ell+1,d-\ell-1)$-triplex, and let the…

Let $d$ be a positive integer, let $k$ be a face dimension, and consider $d$-polytopes with at most $3d-1$ vertices. A minimiser is a polytope attaining the minimum number of $k$-f…

Let $d=d_1+\cdots+d_m$, let $\bbbeta$ be the block-parameter vector, and let $\mathcal{P}_{d,d}^{\bbbeta}$ be the associated block-beta random polytope. All-face-dimensions conject…

Let $d>0$ be odd, and let $\mathcal{T}$ be a triangulation in the remaining case where $f_d$ is odd and $f_d<d$. Spreer–Tobin's small odd-facet vertex maximum conjecture. The true…

Let $\mathcal{T}$ be a triangulation of a closed and connected $d$-dimensional manifold, with $d>0$, $f_0$ vertices, and $f_d$ facets. Spreer–Tobin's sharp vertex bound conjecture.…

Let $d$ be an even integer, and let $\mathcal{T}$ be a (generalised) triangulation of a $d$-manifold $\mathcal{M}$ with $f$-vector $(f_0,f_1,\ldots,f_d)$. Spreer–Tobin's vertex bou…

Klee–Nevo–Novik–Zheng conjecture. Then

Stanley's conjecture. Then

Let $\Sigma$ be a $(d-1)$-dimensional homology sphere, with $g$-vector $g(\Sigma)=(g_0,g_1,\ldots,g_{\lfloor d/2 \rfloor})$. A vector…

Let $(\Delta,\nu)$ be a PL realization of an orientable simplicial $\mathbb{R}$-homology $d$-manifold without boundary in $\mathbb{R}^d$, and let $\Psi(\Delta,\nu)$ be its stress a…

Let $\Delta$ be an orientable simplicial $\Bbbk$-homology $d$-manifold without boundary, and let…

Let $\Delta$ be a simplicial $d$-sphere, or more generally a simplicial homology $d$-sphere. The $h$-vector of $\Delta$ is required to satisfy the Dehn--Sommerville equations, and…

Let $P$ be a $2$-level $d$-polytope, and let $f_0(P)$ and $f_{d-1}(P)$ denote its numbers of vertices and facets, respectively. Vertex-facet product conjecture. For every $2$-level…

Flag Eulerian upper bound conjecture. For every $1\leq i\leq 2m-1$,

Sparse $2d$-vertex polytope nonexistence conjecture. There are no $d$-polytopes with $2d$ vertices and $d^2+d-3$ edges.