Definably isometric cuboids over Hensel minimal fields

Abstract

We show that, over a valued field $K$ with $1$-h-minimal theory $\operatorname{Th}(K)$, two Cartesian products of balls must be equal up to translation and coordinate permutation in order for there to exist a definable surjective isometry between them. For this purpose, we introduce a notion of relative dimension for cuboids in $K^n$ that is computed combinatorially and relates to the notion of dimension for definable sets in Hensel minimal theories when $K$ has infinite residue field.