Cutting a convex body into fat parts and approximating Euclidean distance by graph distances
Abstract
Can one construct a graph G on the set of integer points ℤ2 in the plane such that the length of the shortest path between any two vertices of G differs from their Euclidean distance by at most an absolute constant? This question of Benjamini, Erd\H os, Kleiner, Kozma, Schramm, and the first-named author has been open for a long time. We give an affirmative answer to a weaker form of this question, based on the following geometric statement, which is of independent interest. There exists a constant c>0 such that for every i=1,2,…, every ρ-fat plane convex set S can be cut into 2i convex pieces of equal area, each of which is at least cρ-fat. (A convex set is ρ-fat if the ratio of its inradius to its circumradius is at least ρ.) We prove that there exists an (unweighted) spanning subgraph G of an enlarged copy of ℤ2 such that, for every pair of vertices at Euclidean distance d, their shortest-path distance in G lies between d−O(1) and d+o(d5/6). The same bound can be achieved by a planar graph with vertex set ℤ2, in which every edge joins two vertices at Euclidean distance at most 2.
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BibTeX
@misc{pach2026cutting,
title = {{Cutting a convex body into fat parts and approximating Euclidean distance by graph distances}},
author = {János Pach and Gábor Tardos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20702},
eprint = {2609.20702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20702},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cutting a convex body into fat parts and approximating Euclidean distance by graph distances
AU - János Pach
AU - Gábor Tardos
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20702
UR - https://arxiv.org/abs/2609.20702
AB - Can one construct a graph G on the set of integer points ℤ^2 in the plane such that the length of the shortest path between any two vertices of G differs from their Euclidean distance by at most an absolute constant? This question of Benjamini, Erd\H os, Kleiner, Kozma, Schramm, and the first-named author has been open for a long time. We give an affirmative answer to a weaker form of this question, based on the following geometric statement, which is of independent interest. There exists a constant c>0 such that for every i=1,2,…, every ρ-fat plane convex set S can be cut into 2^i convex pieces of equal area, each of which is at least cρ-fat. (A convex set is ρ-fat if the ratio of its inradius to its circumradius is at least ρ.) We prove that there exists an (unweighted) spanning subgraph G of an enlarged copy of ℤ^2 such that, for every pair of vertices at Euclidean distance d, their shortest-path distance in G lies between d−O(1) and d+o(d^5/6). The same bound can be achieved by a planar graph with vertex set ℤ^2, in which every edge joins two vertices at Euclidean distance at most 2.
ER -