A proof of a conjecture by Haviv, Lyubashevsky and Regev on the second moment of a lattice Voronoi cell

TL;DR

This paper proves a conjecture regarding the second moment of lattice Voronoi cells, establishing a sharp lower bound related to the covering radius and characterizing cases of equality.

Contribution

It provides a rigorous proof of a conjecture on the second moment of lattice Voronoi cells and characterizes the extremal cases.

Findings

Established a sharp lower bound for the second moment

Confirmed the conjecture by Haviv, Lyubashevsky, and Regev

Characterized cases where the bound is attained

Abstract

In this short note we prove a sharp lower bound for the second moment of a lattice Voronoi cell in terms of the respective covering radius. This gives an affirmative answer to a conjecture by Haviv, Lyubashevsky and Regev. We also characterize those lattice Voronoi cells for which this lower bound is attained.