I have some points in 2D space and simply want to fit a line through them (solve for $m$ and $b$ of the equation $y = mx + b$) such that the maximum error for any given point is as small as possible. This is not least-squares linear regression because my goal is not to minimize the total error among all points. Instead, I want to minimize the error of the point that has the worst fit. Is there an equation for how to do this?
In least squares linear regression you are minimizing the L^2 norm of residuals. What you would like to do is to minimize the L^infinity norm. If you do not mind to have an approximate solution, you can minimize the L^p norm for a large p (say p=100, which should be a good approximation of infinity) and do it numerically with standard gradient methods.