Séminaires : Séminaire d'Analyse Fonctionnelle

We show that there exists a convex body that tiles $\R^n$ under translations by $\Z^n$ whose surface area is $\tildeO(\sqrt{n})$. No asymptotic improvement over the trivial $O(n)$ (achieved by the cube $[0,1]^n$) was previously known. A non-convex body achieving surface area $O(\sqrt{n})$ was constructed by Kindler, O’Donnell, Rao, and Wigderson [FOCS 2008].