A Step Beyond Kemperman's Structure Theorem

A classical result of Kemperman gives a complete recursive description of the structure of those subsets $A$ and $B$ of an abelian group that fail to satisfy the triangle inequality, i.e., $\mathrm{\mid }A+B\mathrm{\mid }<\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }$. In this paper, we achieve the complete description in the case when equality holds: $\mathrm{\mid }A+B\mathrm{\mid }=\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }$.