2012/05/28 by Emmanuel Tsukerman, Tsukerman, Emmanuel · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #math.CO
paper · pdf · doi:10.48550/arxiv.1205.6063
Withdrawn - this preprint has been incorporated into arXiv:1303.4139
openalex publication_date 2012/05/28 · arxiv created 2013/09/07 · arxiv updated 2013/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the edge-isoperimetric problem on the graph of the infinite grid ℕ2 in the ℓ∞ metric. We first show that the solutions are not nested, so that techniques other than compressions have to be used. We then show that for any given volume of sets in ℕ2, there exists an optimal set of a specific geometric form and describe this form. We continue on to prove that the optimal perimeter has asymptotic growth rate 2√(7x) as a function of the volume and obtain upper and lower bounds for the optimal perimeter which are within the small additive constant of (35)/(2) of one another, thus effectively solving the discrete isoperimetric inequality on this graph. Finally, we prove that there exist arbitrarily long consecutive values of the volume for which the minimum perimeter is the same.