> This has nothing to do with the coordinates by the way.
I think it does. Both decompose along orthogonal directions. See my comment here https://news.ycombinator.com/item?id=45248881
> This has nothing to do with the coordinates by the way.
I think it does. Both decompose along orthogonal directions. See my comment here https://news.ycombinator.com/item?id=45248881
I mean they decompose in orthogonal components for all Lp norms I think? Is there a norm for which (x,0) is not the closest point to (x,y) on the x-axis?
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
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.
I think that's arguably an a posteriori explanation: you can find orthogonal coordinates with respect to which the L^2 norm has a nice form, but you can also single out the L^2 norm in various ways (for example, by its large symmetry group, or the fact that it obeys the parallelogram law—or even just the fact that "orthogonal" makes sense!) without ever directly referencing coordinates.