Curve through two points

Consider values along a horizontal x-axis ranging from 1 to 100. Suppose at x=1, the y value is given to be y₁=1. Likewise at x=100 the y value is given to be y₁₀₀=100. Now we are to find all the intervening points, $y_2,y_3,\ldots,y_{99}$.Let us use the L₁ criterion  

$$\min_{y_2,y_3,\ldots,y_{99}} \quad \vert y_2-y_1\vert + \vert y_3-y_2\vert + \vert y_4-y_3\vert + \cdots$$ (7)

The solution to this problem is any curve with a positive slope since all such curves result in the same value of 99 for (7) That is quite a lot of curves!

Now we begin to appreciate the strange flavor of L₁. We appreciate the idea that solutions are ``intervals''. But it is distressing to realize that we could often have graphical difficulty displaying the results. In practice we might need to settle for ``seeing some examples.'' Perhaps a satisfactory way of generating those examples would be by using random starting values for the fitting.

Suppose we set up the Busch problem with L₁. Perhaps we will find the solution is not a unique surface. It might turn out to be a ``mat'' of variable thickness. It would be annoying to try to display the thickness, but perhaps the thickness is related to the uncertainty of the result. That should have value.