Unit-Norm Vectors under Different p-Norms

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This Demonstration shows how unit-norm vectors look under different -norms, which are standard norms for finite-dimensional spaces.


In mathematics, a norm is a function that assigns a length (or size) to a vector. The vector is an object in a vector space, and can thus be a function, matrix, sequence, and so on. A -norm is a norm on a finite-dimensional space of dimension defined as


This Demonstration shows sets of unit-norm vectors for different -norms.

The norm for is called the Manhattan or taxicab norm because represents the driving distance from the origin to following a rectangular street grid

The norm for is the usual Euclidean square norm obtained using the Pythagorean theorem

The norm for is simply the maximum over and ,

Vectors ending on the red lines are of unit norm in the corresponding -norm.


Contributed by: Jelena Kovacevic (June 2012)
Open content licensed under CC BY-NC-SA




[1] M. Vetterli, J. Kovačević, and V. K. Goyal, Foundations of Signal Processing, Cambridge: Cambridge University Press, 2014. www.fourierandwavelets.org.

[2] Wikipedia. "Norm." (Jun 12, 2012) en.wikipedia.org/wiki/Norm_%28 mathematics %29.

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