How does the shape of a ball change as we measure distance differently?
Image: Created by Wolfgang Beyer with the program Ultra Fractal 3., CC BY-SA 3.0, via Wikimedia Commons
the Lp norm ball shape changes as p goes from 1 to 2 to infinity
How does the shape of a ball change as we measure distance differently?
Imagine you're measuring how far apart two friends are standing. If you use a simple ruler, you're measuring straight-line distance. But what if you want to consider how far apart they are in a more complex way, like walking through a park?
The Lp norm is like choosing different ways to measure distance. With p=1, it's like walking straight through obstacles. As p increases, it's like considering the longest path, even if it's winding. The ball shape changes from a diamond to a circle to a square as p goes from 1 to 2 to infinity.
Example
If your friends are 3 meters apart straight-line (p=1), they might be 5 meters apart if you consider the winding path through the park (p=2), and 10 meters apart if you consider the longest possible route (p=infinity).
Remember this
The Lp norm measures distance in different ways, changing the shape of the ball that represents all possible distances.
Text adapted from Wikipedia, licensed under CC BY-SA 4.0.
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the L1 unit ball is a diamond shape and the L2 unit ball is a circle
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Norm (mathematics)
L∞ norm equals max absolute value
the L1 norm is not differentiable at zero
Absolute value's kink makes it non-differentiable at zero
Minkowski spacetime
Minkowski distance formula: D = (Σ |x_i - y_i|^p)^(1/p)
Euclidean geometry
Euclidean distance measures absolute position in space
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