Given $n$ points $(x_i, y_i)$ in the unit square with $x_i = {{i - 1} \over {n - 1}}$ uniformly spaced and $0 \leq y_i \leq 1$, consider the best-fit L1 line and the best-fit L2 line:
$ \qquad$ L1 aka
Least absolute deviation line:
minimize $\sum |y_i - a_1 \, x_i - b_1|$
$ \qquad$ L2 aka least squares or
simple linear regression line:
minimize $\sum (y_i - a_2 \, x_i - b_2)^2 $
How far apart can these two lines be, over all sets of $n$ points $(x_i, y_i)$ as above ?
The following patterns of $y_i \in \{0, 1\}$ have L1 lines constant 0, which simplifies life. They are local maxima, > nearby patterns, but that doesn't prove anything.
