42 problems

Let $\mathcal{F}\subseteq\binom{[n]}{k}$, with $k>d$. A $d$-simplex is a collection of $d+1$ sets with empty common intersection while every proper subcollection has nonempty inter…

Corsten–Frankl's conjecture. A simplex is diameter-Ramsey if and only if its circumcenter belongs to its convex hull.

Let $S$ be a hyperbolic or spherical $n$-simplex, and let its volume, viewed as a function of the dihedral angles, be continuously extended to degenerated simplexes. Let the closur…

Let $S$ be a hyperbolic or spherical $n$-simplex, and regard its volume as a function of its dihedral angles. The space of such simplexes may degenerate, yielding a closure of the…

Unique-centre conjecture. If a simplex has a unique simplex centre, then it is equifacetal.

Integer-coordinate conjecture. Every Heron simplex can be represented with integer coordinates.

Let $K\subset\mathbb{R}^n$ be a convex body, let $-K=\{-x:x\in K\}$ be its reflection through the origin, let $\operatorname{vol}_n$ denote $n$-dimensional volume, and let $A+B$ de…

Let $ssim(n,k,k-1)$ denote the maximum size of a family of $k$-subsets of $[n]$ containing no special $(k-1)$-simplex. Frankl–Füredi conjecture. For $n\geq 2k$, … The bound is att…

Let $ssim(n,k,d)$ denote the maximum size of a family of $k$-subsets of $[n]$ containing no special $d$-simplex. Frankl–Füredi conjecture. For $k=d+1$ and $k=d+2$, … The source no…

Let $\mathcal{F}\subseteq\binom{[n]}{k}$, with $k\geq 3$. A triangle is a $2$-simplex, namely three sets with empty total intersection but every pair having nonempty intersection.…

Let $N$ be a prime with $N\geq 3$. A spanning simplex is a lattice simplex whose lattice points at height one generate the ambient lattice, and a non-degenerate thin code is a thin…

A thin simplex is a lattice simplex of lattice width at most one, and its lattice width is the minimum width over all nonzero integral linear functionals. Width-one conjecture. In…

A thin simplex is a lattice simplex of lattice width at most one. Finiteness conjecture. In each even dimension there are only finitely many thin simplices of lattice width…

Sallee's conjecture. Only the regular simplex has maximal width among all simplices inscribed in a sphere in $\mathbb{R}^n$.

Let $\Delta^n=\{(x_1,\ldots,x_n)\in[0,1]^n: x_1\leq\cdots\leq x_n\}$, and let $F\subseteq\Delta^n$. Write $F^c=\Delta^n\setminus F$. The rank of a set is defined by the preceding r…

Let $\mathrm{DIAM}(d,k,n)$ be the maximum number of unit $k$-simplices spanned by an $n$-point set in $\mathbb{R}^d$ of diameter $1$. Conjecture on unit simplices in diameter graph…

Let $d\geq 4$, let $k\leq d$, and let $\mathrm{C}(d,k,n)$ denote the maximum, over all $k$-vertex simplices $S$ and all $n$-element sets $P\subseteq\mathbb{R}^d$, of the number of…

Let $Q_n$ be the $n$-dimensional cube. A simplex $S\subset Q_n$ is extremal when $\xi(S)=\xi_n$, and it is equisecting when the $(n-1)$-dimensional hyperplanes containing its faces…

Let ${\mathcal P}\subset\mathbb{R}^d$ be a $d$-dimensional simplex, and let its solid angle be measured in the usual way. Karasev's smallest-solid-angle conjecture. Every $d$-dimen…

Conjecture d'Antunes–Freitas. La fonction d'écart $\xi:\mathfrak{M}_n\to\mathbb{R}$ est propre, et le simplexe régulier, défini par des sommets $p_0,p_1,\ldots,p_n\in\mathbb{R}^n$…

A family of $d$-simplices in $\mathbb{R}^d$ is neighbourly if the common part of every two members is a $(d-1)$-dimensional set. Let $c_d$ denote the maximum cardinality of a neigh…

Let $\omega\in\mathbb{R}^{d+1}_{>0}$ satisfy $\omega_0\leq\omega_1\leq\ldots\leq\omega_d$, and let $j\in\{1,\ldots,d\}$. Let $S(\omega)$ be the weighted simplex and let $\sigma_r$…

Let $S(\mathbf{1}_{n+1})$ be the standard terminal simplex, and let $\mu_d$ denote the $d$-th covering minimum. The covering-minima conjecture. For every $n \in \mathbb{N}$ and…

For a positive integer $r$ and a positive integer dimension $d$, define $\boldsymbol{q}_r=(r-1,(r-1)r,(r-1)r^2,\dots,(r-1)r^{d-1})$ and let the base-$r$ $d$-simplex be … Here…

Conjecture. The four-dimensional absorption index satisfies