Isoperimetric stability in lattices Article Swipe

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Isoperimetric inequality Combinatorics Boundary (topology) Vertex (graph theory) Conical surface Mathematics Cayley graph Digraph Discrete mathematics Geometry Mathematical analysis Graph

Ben Barber , Joshua Erde , Peter Keevash , Alexander Roberts ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.5817/cz.muni.eurocomb23-015 · OA: W4380876777

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 $B$.