Here is a geometric interpretation of the solutions for the equation in c above. We use the alternate form of Lasso constraints 
.
The Lasso constraint take the form , which when plotted take the familiar shape of a diamond centered at origin )
. Next consider the squared optimization constraint %5E2%20%2B%20(y_2%20-%20%5Chat%7B%5Cbeta%7D_1x_%7B21%7D%20-%20%5Chat%7B%5Cbeta%7D_2x_%7B22%7D)%5E2)
. We use the facts 
, 
, 
, 
and 
to simplify it to
Minimize: x_%7B11%7D)%5E2)
.
This optimization problem has a simple solution: 
. This is a line parallel to the edge of Lasso-diamond 
. Now solutions to the original Lasso optimization problem are contours of the function x_%7B11%7D)%5E2)
that touch the Lasso-diamond . Finally, as 
and 
very along the line , these contours touch the Lasso-diamond edge at different points. As a result, the entire edge is a potential solution to the Lasso optimization problem!
Similar argument can be made for the opposite Lasso-diamond edge: 
.
Thus, the Lasso problem does not have a unique solution. The general form of solution is given by two line segments:
and
