What question did this study set out to answer?

The aim is to establish lattice covering bounds for a broader class of convex bodies beyond symmetry.

January 24, 2026

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

Key Points

The aim is to establish lattice covering bounds for a broader class of convex bodies beyond symmetry.
Analysis of lattice covering densities in various convex bodies
Extension of existing theorems to non-symmetric cases
Examination of anti-blocking and locally anti-blocking bodies
Established a generalized covering density bound for anti-blocking bodies
Showed that this bound holds for polytopes with vertices
Improved understanding of lattice coverings in non-symmetric convex bodies

Abstract

Abstract In 2021, Ordentlich, Regev, and Weiss made a breakthrough that the lattice covering density of any ‐dimensional convex body is upper bounded by , improving on the best previous bound established by Rogers in 1959. However, for the Euclidean ball, Rogers obtained the better upper bound , and this result was extended to certain symmetric convex bodies by Gritzmann. The constant above is independent on . 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 ‐dimensional polytopes with vertices.

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