On Sunday, July 7, 2013 6:43:41 PM UTC-7, quasi wrote: > To show that, assume an optimal line L with slope a passes through > the point (x_k,y_k). > > By the point-slope formula, the line has the equation ... > y = a*x + (y_k - a*x_k) > > Let s = SUM(y_i - y(x_i)). > > Since L is an optimal bounding line, s is minimal, hence a change > in the value of a, if it doesn't break the bounding condition, > cannot decrease the sum. > > But the sum s is at most linear as a function of a. > Case (1): s is degree 1 as function of a. > If L does not pass through any of the other N-1 points, a > sufficiently small positive or negative change in a will decrease > the value of s
This is not correct. A small change in a can either increase or decrease s, depending on the values x_i. Whether L passes through any of the other points is irrelevant.