p-norm ball
p norm un a vector space
A norm in a vector space is a function that asign a positive real number to each vector, is positive-scalar multiplicative and satisfies the triangular inequality.
1. for every and
2. for every and every scalar
3. for every pair
Each norm naturaly define a distance that makes the vector space a metric space.
The distnace is defined by
In the plane R2 we can obtain a very well studied group of norms, named the p-norms.
Consider , and define for
We define the p-norm ball as the set of all vectors in such that de p-distance to the origin is less than 1.
In the following animation we can see the plot of the border of the p-ball for some values of p moving from 1 to 4.