TallKite

You can see these distances are associated with different L norms. The L₁ norm and L∞ norms are each others' duals and the L₂ norm is self-dual. Here's one way to see that relationship: the L₁ norms times the L∞ norms give the same answer as the L₂ norms times themselves.  

 

These come up in tuning. When you minimize the L∞ norm on the prime error, this causes a minimization of the L1 norm on interval error (that's Paul's "minimax", because the L∞ norm of a vector is simply the max value of any of its entries; I understand it that way because your "king" can move as diagonally as necessary, and so he'll just move diagonally in every dimension until he runs out of dimensions he needs to go except for one, at which point he continues straight along that dimension). And if you minimize the L1 norm on the prime error, this causes a minimization of the L∞ norm on interval error. So if you wanted to use L∞ norm for interval error, you'd set your tuning optimizer to minimize the sum of the absolute values of errors per prime. If you have any questions, let me know -- I'm not rock solid on this stuff yet, but I think it's pretty interesting. Dave and I have attempted to improve our geometric intuition for dual norms' effects on tuning, but it's been a while since I looked at it.