[srean]

Not quite sure what you mean but now I agree with your previous point that the coordinate system is irrelevant because these normal derived Lp metrics can be specified in a coordinate free way just by specifying their unit ball set

[shiandow]

Wel what does it mean to be orthogonal? I'd argue that if you only have a metric then a line (segment) is orthogonal to some set if it is the shortest path from a point to a set. For norms it should work to only consider straight lines.

What I was getting at is that the x and y grid is orthogonal for all Lp norms. The interesting stuff happens if you pick a different line as x-axis. Orthogonal projection to a different axis still works though, the resulting grid just won't be orthogonal in the conventional sense.