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.[br][math]N:V\longrightarrow\mathbb{R}[/math][br]1. [math]N\left(v\right)\ge0[/math] for every [math]v\in V[/math] and [math]N\left(v\right)=0\Leftrightarrow v=0[/math][br]2. [math]N\left(kv\right)=|k|N\left(v\right)[/math] for every [math]v\in V[/math] and every scalar [math]k[/math][br]3. [math]N\left(v+w\right)\le N\left(v\right)+N\left(w\right)[/math] for every pair [math]v,w\in V[/math][br][br]Each norm naturaly define a distance that makes the vector space a metric space.[br]The distnace is defined by[br][math]d\left(v,w\right)=N\left(v-w\right)[/math][br][br]In the plane R2 we can obtain a very well studied group of norms, named the p-norms.[br][br]Consider [math]1\le p<\infty[/math], and define for [math]v=\left(x,y\right)\in\mathbb{R}\times\mathbb{R}[/math][br][math]\parallel v\parallel_p=\parallel\left(x,y\right)\parallel_p=\left(x^p+y^p\right)^{\left(\frac{1}{p}\right)}[/math][br][br]We define the p-norm ball as the set of all vectors in [math]\mathbb{R}\times\mathbb{R}[/math] such that de p-distance to the origin is less than 1.[br][br][math]B_p=\left\{v\in\mathbb{R}^2:\parallel v\parallel_p\le1\right\}[/math][br][br]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.

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