I know unit ball for $p$-norm with $p = 2$ is a square, my confusion is how does it look like in $\mathbb{R}^3$ space.
In $\mathbb{R}^3$ space it looks like a cuboid, is this correct ?
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You mean $p=1$? – Augustin Jan 29 '16 at 10:09
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its $$p = 1, 2, \infty$$ – Nithish Jan 29 '16 at 10:10
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1If $p=2$, it looks like a circle. It looks like a square if and only if $p=1$ or $p=\infty$. – Augustin Jan 29 '16 at 10:11
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even in R3 space too ? – Nithish Jan 29 '16 at 10:12
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1In $\mathbb{R}^3$, for $p=2$ it's a regular ball. For $p=1$ it is indeed a cuboid (the vertices are the points of the axis with abcissae $-1$ or $1$). – Augustin Jan 29 '16 at 10:14
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Thank you for clearing my query – Nithish Jan 29 '16 at 10:15
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Let us continue this discussion in chat. – Nithish Jan 29 '16 at 10:16
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The unit ball of $(\mathbb{R}^n,\|\cdot\|_p)$ is $$B_p=\{x\in\mathbb{R}^n, \|x\|_p\leq 1\}.$$
So, if $p=\infty$, then
$$B_\infty=\{x\in\mathbb{R}^n, \forall 1\leq i\leq n, |x_i|\leq 1\}=[-1,1]^n,$$
which is a square in $\mathbb{R}^2$, a cube in $\mathbb{R}^3$, and is probably named hypercube in general.
If $p=2$, then
$$B_2=\{x\in\mathbb{R}^n, \sum_{i=1}^n(x_i)^2\leq 1\},$$
which is a disk in $\mathbb{R}^2$, a round ball in $\mathbb{R}^3$ (like the earth, in first approximation!), and is usually called a ball (or a round ball) in higher dimension, as by default, this is the most usual norm.
If $p=1$, then
$$B_1=\{x\in\mathbb{R}^n, \sum_{i=1}^n|x_i|\leq 1\},$$
which is a square in $\mathbb{R}^2$ (but different from $B_\infty$), and in $\mathbb{R}^3$ it is not a cube, but an octahedron (use google image), whose corners are the points $(\pm 1,0,0), (0,\pm 1,0), (0,0,\pm 1)$. I don't know if it has a name in higher dimension, but one piece of its boundary, namely $\Delta=\{x\in\mathbb{R}_+^n, \sum_{i=1}^nx_i= 1\},$ is called a simplex (which is a segment if $n=2$, and a triangle if $n=3$)
John Steinbeck
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