TITLE: For which norm-Euclidean domains is the proof interval less than $1$?
QUESTION [9 upvotes]: I saw a proof somewhere (can't find it at the moment) that shows $\textbf{Z}[\sqrt{-2}]$ is norm-Euclidean because for any pair of nonzero numbers $a, b$ it's possible to find a remainder $r$ such that $$N(r) \leq \frac{3}{4} N(b).$$ In Bolker's error-prone book, a proof covering $\textbf{Z}[\sqrt{-2}]$, $\textbf{Z}[i]$, $\textbf{Z}[\sqrt{2}]$ shows that in each of these domains it's always possible to find $$N(r) \leq \frac{1}{2} N(b).$$ Maybe I'm recalling incorrectly, because it seems to me that if $\frac{1}{2}$ does it for these four domains, then $\frac{3}{4}$ is too much for $\textbf{Z}[\sqrt{-2}]$. Basic logic suggests that at the most general level, $1$ is the absolute maximum, with strict inequality, of course.
Do any of the norm-Euclidean quadratic rings require $1$ for the proof interval?  Or is $\frac{1}{2}$ always sufficient? Or is there more variety than this?

REPLY [4 votes]: The infimum of all values of $c$ such that $f(r) \le c f(b)$ for all pairs of $r$ and $b$ is called the Euclidean minimum for the (Euclidean) function $f$. If $f$ is the absolute value of the norm, the classical name is 
"inhomogeneous minimum of the norm form". 
If $f$ is the minimal Euclidean function on a domain as defined by Motzkin, then $c = 1$. If $f$ is the absolute value of the norm, then we must have $c < 1$ if the ring is norm-Euclidean, as was shown by Barnes and Swinnerton-Dyer in the quadratic and by Cerri in the case of general number fields.
If the ring contains a principal ideal of norm $a > 1$, then we always have $c > 1/a$. I do not know whether every rational value of $c > 0$ does occur
as a Euclidean minimum for the norm function.
For more information, see my survey.