Ehrhart-equivalent $\mathbf{3}$-polytopes are equidecomposable

We show that if two lattice $3$-polytopes $P$ and $P^{\mathrm{\prime }}$ have the same Ehrhart function then they are ${\mathrm{GL}}_3(\mathbb{Z})$-equidecomposable; that is, they can be partitioned into relatively open simplices $U_1,\dots, U_k$and$U’_1,\dots,U’_k$suchthat$U_i$and$U’_i$ are unimodularly equivalent, for each $i$.