Thanks for writing this. I love the illustration of the concept with the video. I have little knowledge or background in the area so it is very useful.
I got confused where you introduced the objective function though. Is the reader supposed to understand where that comes from or are you just presenting it as a fact? It also doesn't make sense to me that the function is minimised over 'b' but 'b' does not appear in the function?
The minimization function is indeed "as a fact"; in the sense that it is (somewhat) the formulation of Cortes and Vapnik (http://link.springer.com/article/10.1007%2FBF00994018), which is the main formulation for Support Vector Machines.
The value of 'b' is minimized and the value if found in the first constraint. The value "b\||w||" determines the offset of the hyperplane to the origin and 'w' is the normal vector of that plane.
I got confused where you introduced the objective function though. Is the reader supposed to understand where that comes from or are you just presenting it as a fact? It also doesn't make sense to me that the function is minimised over 'b' but 'b' does not appear in the function?