On lattice coverings by locally anti-blocking bodies and polytopes with few vertices

In 2021, Ordentlich, Regev and Weiss made a breakthrough that the lattice covering density of any $n$-dimensional convex body is upper bounded by $cn^{2}$, improving on the best previous bound established by Rogers in 1959. However, for the Euclidean ball, Rogers obtained the better upper bound $n(\log_{e}n)^{c}$, and this result was extended to certain symmetric convex bodies by Gritzmann. The constant $c$ above is independent on $n$. In this paper, we show that such a bound can be achieved for more general classes of convex bodies without symmetry, including anti-blocking bodies, locally anti-blocking bodies and $n$-dimensional polytopes with $n+2$ vertices.