5 problems

Let a space-filling polyhedron be a convex polyhedron that tiles three-dimensional space by translations, and restrict to such polyhedra of unit volume. The truncated octahedron is…

Let $P$ be a parallelohedron in $\mathbb R^3$ with unit inradius. Bezdek's Rhombic Dodecahedral Conjecture. The surface area of $P$ is at least … which is the surface area of the r…

Let $P$ be a three-dimensional parallelohedron of unit volume, and let $\operatorname{surf}(P)$ denote its surface area. Regular truncated-octahedron conjecture. Among three-dimens…

Strong dimension conjecture. For every dual $k$-cell $D$ there is a $k$-dimensional lattice $\Lambda$ such that there is a $k$-dimensional cell in the Delone tessellation of…

Dimension conjecture. For every dual $k$-cell, the dimension of its affine span is equal to $k$.