A circle is made up of points equidistant from the center. But what does “equidistant” mean? Measuring distance implies a value judgment — for example, that moving to the left is just the same as moving to the right, moving forward is just as hard as moving back.

But what if you’re on a hill? Then the amount of force to go uphill is different than the amount to go downhill. If you drew a picture of all the points you could reach with a fixed amount of work (equiforce or equiwork or equi-effort curve) then it would look different — slanted, tilted, bowed — but still be “even” in the same sense that a circle is.

Here’re some brain-wrinkling pictures of “circles”, under different L_p metrics:

astroid p=⅔
p = ⅔

The subadditive “triangle inequality” A→B→C > A→C no longer holds when p<1.

p = 4p = 4 

 p = 1/2
= ½
. (Think about a Poincaré disk to see how these pointy astroids can be “circles”.)
 p = 3/2 p = 3/2 

 workin on my ♘ ♞ movesThe moves available to a knight ♘ ♞ in chess are a circle under L1 metric over a discrete 2-D space.


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