The converse of a theorem by Bayer and Stillman

Bayer–Stillman showed that reg(I)=reg(ginτ(I))reg(I)=reg(ginτ(I)) when τ is the graded reverse lexicographic order. We show that the reverse lexicographic order is the unique monomial order τ   satisfying reg(I)=reg(ginτ(I))reg(I)=reg(ginτ(I)) for all ideals I  . We also show that if ginτ1(I)=ginτ2(I)ginτ1(I)=ginτ2(I) for all I  , then τ1=τ2τ1=τ2.