On two counterexamples in the geometry of numbers

Abstract

We give counterexamples to two optimization problems in dimensions eight and nine. 1. The Cartesian-product problem posed by Cassels for critical determinants and later formulated by Zong for lattice packings and for packings allowing translations but not rotations: whether the corresponding product inequalities are always equalities. 2. A question raised by Sarnak and formulated as a conjecture in Chiu: whether, among unit-volume flat tori, height is minimized by a lattice maximizing the length of its shortest nonzero vector. The first counterexample is exact and also disproves the natural product formula for unrestricted congruent packings. The second is numerical but within reasonable floating-point accuracy.