11 problems

Nonexistence conjecture. For $n>1$, there are no centrally symmetric perfect Delaunay polytopes whose translates tile $\mathbb{R}^n$.

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…

Growth conjecture. The growth of the number of perfect Delaunay polytopes in dimension $d$ up to affine equivalence is not prohibitive.

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

Conway–Sloane's conjecture. The function $f$ characterizes $L$ up to conjugacy in every dimension; Conway and Sloane proved this in dimensions at most $3$.

Let $Erdahl(n)$ denote the space used in the paper, and for $f\in Erdahl(n)$ let $proj(f)=g$, where $(g,h)$ is the unique extremal pair associated with $f$. Continuity conjecture f…

For each $n\geq 6$, let $LD_n$ be the lattice defined in Dutour (2005), and let a covering maximum be a lattice whose covering density decreases under perturbation. The covering-de…

Face-number conjecture. For every $d$-dimensional perfect prismatoid $P$, there exists a $d$-dimensional lattice Delaunay polytope $Q$ such that

Let $P$ be an $n$-dimensional polytope whose vertices belong to a lattice $L$, and suppose that its interior contains no lattice point. The lamination-number conjecture. The lamina…

Let $P$ and $P_1$ be perfect Delaunay polytopes for $\mathbb{Z}^n$ and call them adjacent when the quadratic rank of their common vertex set is $2$. An adjacency component is a col…

Let $P$ be a perfect Delaunay polytope, meaning a lattice polytope circumscribed by a unique possibly degenerate ellipsoid. Basicity conjecture. Every perfect Delaunay polytope is…