On the packing densities and the covering densities of the Cartesian products of convex bodies (Q1780917)

For a convex body \(K \subseteq {\mathbb R}^n\), let \(\delta (K)\) denote its maximal translative packing density, \(\delta^* (K)\) its maximal lattice packing density, \(\theta (K)\) its minimal translative covering density, and \(\theta^* (K)\) its minimal lattice covering density. The paper under review proves the following inequalities: (1) If \(K_1\) is a regular convex body, then for all convex bodies \(K_2\), \[ \delta^* (K_1 \oplus K_2) <\delta (K_2). \] (2) For every convex body \(K_1\) that is not a tile, there is a convex body \(K_2\) such that \[ \theta (K_1 \oplus K_2) <\theta (K_1) \cdot \theta (K_2). \] (3) For every convex body \(K_1\) that is not a tile, there is a convex body \(K_2\) such that \[ \theta^* (K_1 \oplus K_2) <\theta^* (K_1) \cdot \theta^* (K_2). \] (4) If \(K_1\) is a regular convex body, then for all convex bodies \(K_2\), \[ \theta^* (K_1 \oplus K_2) > \theta (K_2). \] These results are motivated by the question whether \[ \delta (K_1 \oplus K_2) = \delta (K_1) \cdot \delta (K_2) \quad \text{and} \quad \delta^* (K_1 \oplus K_2) = \delta^* (K_1) \cdot \delta^* (K_2) \] hold for all convex bodies \(K_1\) and \(K_2\). The author also offers a combinatorial version of this problem.