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

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 Kn,dBM based on finite subsets of Zn theoretically. In this work, we present an algorithm to construct ε-nets for K2,dBM and a (1/4)-net for C2,dBM is constructed. To the best of our knowledge, this is the first concrete ε-net for C2,dBM for such a small ε.