Efficient construction/description of low-surface-area tiling bodies

Ascertain whether there exist methods to efficiently describe or construct, for each n, a convex body K ⊂ ℝ^n whose integer translates tile ℝ^n, with vol_n(K)=1 and surface area vol_{n−1}(∂K)=n^{1/2+o(1)}, for example via a constant-factor polynomial-time optimization oracle or another comparably efficient representation, despite the fact that any such K cannot be represented as the intersection of n^{O(1)} half-spaces.

Background

Recent work [NR23] constructs, for every n, a convex polytope K whose integer translates tile ℝ^n and whose surface area satisfies vol_{n−1}(∂K)=n^{1/2+o(1)}. This is equivalent to an isoperimetric quotient iq(K)=n^{1/2+o(1)} under the tiling condition vol_n(K)=1.

For potential algorithmic applications, an efficient description (e.g., polynomial-time optimization oracle) of such a tiling body would be needed. The paper’s Theorem 1 implies that the number of facets of any such K must grow faster than any power of n, ruling out a short half-space description as an efficient representation.

The authors therefore single out as open whether other routes to an efficient construction or description of such tiling bodies exist.