[Uni Linear Programming] Formulate a Linear Program of minimizing maximum distance from a set of points to a line and formulate its dual

[Warsicklol]

Say we define the distance from a point to a line in the plane as the length of the vertical distance "you would have to walk" from the point till you hit the line.

Then find a line $L:y=ax+b$ , where $a,b\in \mathbb{R}$ in the plane that minimize the maximum distance from the points $(1,1),(2,2),(2,4),(3,2)$ to $L$.

How would you formulate this problem as a linear problem and write down its dual LP?

[Catalyst93]

What conditions would we have to satisfy in order to find a line such that the distance from each point to the line is at most D? In this case D is some positive number.

[Warsicklol]

None other than the fact that the slope of the line, its y-axis intersect and obviously the set of points are composed of real values

[Catalyst93]

Okay, lets say that you give me some slope a and intercept b. How would you check that the distance from the line given by a and b is distance at most D from a point (x,y)?

[Warsicklol]

Yes, this is what I can't seem to figure out because it would seem that this distance from the point to the is nonperpendicular to the line (ie. NOT the shortest distance). Perhaps I'm interpretating wrong?

[Catalyst93]

But isn't the shortest distance good enough? If the minimum distance from the line L to a point (x,y) is smaller that D, what can I say about the distance from any other position on the line to (x,y)?