18 problems

Characteristic-polynomial conjecture. All roots of $f(x)$ are real numbers. If $m=n+2$, the points are in general position in $H^n$, and exactly $s$ of the $\alpha_i$ are positive,…

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 $N$ 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…

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 ${\mathcal P}\subset\mathbb{R}^d$ 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…

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…

Diameter-Ramsey simplex conjecture. A simplex is diameter-Ramsey if and only if its circumcentre is contained in its convex hull.

Let $\mathbf{u}_0,\dots,\mathbf{u}_n$ be unit vectors representing the vertices of a simplex in complex projective space, in general position, and let … Let $d_{\min}$ denote the m…

Let $n$ be a positive integer, and let a simplex and its inverted simplex be given in $n$-dimensional space. Consider their Minkowski interpolations and define the volume ratio as…

Weak Spread Out Simplices Conjecture. Any $n$ spread out simplices in $n\Delta_{d-1}$ can be achieved as the simplices of an acyclic system of permutations.

Acyclic System Conjecture. Any acyclic system of permutations on the edges of the simplex $n\Delta_{d-1}$ is achievable as the system of permutations of a fine mixed subdivision.

Let $\mathfrak{M}_n$ be the moduli space of all $n$-simplices with unit diameter. For $n\geq 2$, let $p_0,p_1,\ldots,p_n\in\mathbb{R}^n$ satisfy … so that these points define a reg…

For integers $d\geq 3$ and $k\geq 2$, a $d$-dimensional $k$-reptile simplex is a $d$-dimensional simplex that can be tiled without overlaps by $k$ smaller simplices congruent and s…