Isoperimetric stability in lattices

Ben Barber, Joshua Erde, Peter Keevash, Alexander Roberts

Research output: Working paper


We obtain isoperimetric stability theorems for general Cayley digraphs on $\mathbb{Z}^d$. For any fixed $B$ that generates $\mathbb{Z}^d$ over $\mathbb{Z}$, we characterise the approximate structure of large sets $A$ that are approximately isoperimetric in the Cayley digraph of $B$: we show that $A$ must be close to a set of the form $kZ \cap \mathbb{Z}^d$, where for the vertex boundary $Z$ is the conical hull of $B$, and for the edge boundary $Z$ is the zonotope generated by $B$.
Original languageEnglish
Number of pages20
Publication statusPublished - 28 Jul 2020


  • Isoperimetry
  • Stability
  • Convex Geometry


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