Construction of ε -Nets for the Space of Planar Convex Bodies Endowed with the Banach–Mazur Metric

Yanmei Chen, Yunfang Lyu, Shenghua Gao and Senlin Wu ()
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Yanmei Chen: School of Mathematics, North University of China, Taiyuan 030051, China
Yunfang Lyu: Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China
Shenghua Gao: Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China
Senlin Wu: School of Mathematics, North University of China, Taiyuan 030051, China

Mathematics, 2025, vol. 13, issue 8, 1-12

Abstract: In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of ε -nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et al. provided a possible way of constructing ε -nets for K n , d B M based on finite subsets of Z n theoretically. In this work, we present an algorithm to construct ε -nets for K 2 , d B M and a ( 1 / 4 ) -net for C 2 , d B M is constructed. To the best of our knowledge, this is the first concrete ε -net for C 2 , d B M for such a small ε .

Keywords: Hadwiger’s covering problem; Banach–Mazur metric; ε -net; planar convex body; enumeration of convex lattice set (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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