A General Method to Determine Limiting Optimal Shapes for Edge-Isoperimetric Inequalities

SMC Author

Ellen Veomett

SMC Affiliated Work

1

Status

Faculty

School

School of Science

Department

Math/Computer Science

Document Type

Article

Publication Date

2-17-2017

Publication / Conference / Sponsorship

The Electronic Journal of Combinatorics

Description/Abstract

For a general family of graphs on Zn, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in Rn. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality and Minkowski's theorem on Mixed Volumes. This translation allows us to conclude, under a reasonable assumption about the discrete problem, that the shapes of the optimal sets in the discrete problem approach the shape of the optimal set in the continuous problem as the size of the set grows. The solution is the zonotope defined as the Minkowski sum of the edges of the original graph.
We demonstrate the efficacy of this method by revisiting some previously solved classical edge-isoperimetric problems. We then apply our method to some discrete isoperimetric problems which had not previously been solved. The complexity of those solutions suggest that it would be quite difficult to find them using discrete methods only.

Keywords

Discrete isoperimetric problem, Continuous isoperimetric problem, Infinite graphs, Lattices

Scholarly

yes

Volume

24

Issue

1

First Page

#P1.26

Disciplines

Computer Sciences | Mathematics

Original Citation

Veomett, Ellen. “A General Method to Determine Limiting Optimal Shapes for Edge-Isoperimetric Inequalities.” With E. Tsukerman. The Electronic Journal of Combinatorics, 24(1) (2017), #P1.26.

Link to Full Text

Share

COinS