[bowaggoner]

You're right and we're on the same page about the solution. The reasoning behind my comment of "interesting" is that formula you gave is not well-defined for p=infinity, although we can define it as the limit of that expression as p-->infinity. Furthermore, for p < infinity the lp-norm can assign a well-defined penalty to each x, and the goal is to minimize the sum of the penalties. But that's not really true for p=infinity.

[FabHK]

In the limit it's indeed the max of the absolute differences, so that's how you conventionally define the infinity norm (even though the base formula itself is not well defined, just as with 0). And when you try to minimise the max of the distances, you get the midrange.