We study the minimal possible density \(\delta (K)\) of a non-separable lattice of translates of a given convex body K. We obtain two main results. First, we completely resolve Endre Makai’s conjecture: a two-dimensional convex body K satisfies the inequality \(\delta (K)\le \frac{\pi \sqrt{3}}{8}\) , where equality is attained if and only if K is an ellipse. Second, we obtain a new bound in the three-dimensional case: every three-dimensional convex body K satisfies the inequality \(\delta (K)\le \frac{\pi }{4\sqrt{3}}\) . This inequality is proven using the celebrated Petty projection inequality.
