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Bayer-Stillman showed that $reg(I) = reg(gin_\tau(I))$ when $\tau$ is thegraded reverse lexicographic order. We show that the reverse lexicographicorder is the unique monomial order $\tau$ satisfying $reg(I) =reg(gin_\tau(I))$ for all ideals $I$. We also show that if $gin_{\tau_1}(I) =gin_{\tau_2}(I)$ for all $I$, then $\tau_1 = \tau_2$.

The converse of a theorem by Bayer and Stillman