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I read about half space and hyper plane and keep getting confused about which is which and how people are using it. I would really appreciate if somebody can give me an example in simple language over the math one written on wikipedia.Half Space

I made a mistake of writing half planes instead of hyper plane. Corrected it.

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Half space = half plane in dimension 2. –  Damien L Feb 12 '13 at 16:22
    
In a word, the difference is the difference between “$\ge$” and “$=$”. –  Lubin Feb 12 '13 at 17:24

3 Answers 3

A hyperplane is a subset of a Euclidean space of one less dimension than the whole space. As such, it is defined by one linear equation. In $\mathbb R^3$, the plane created by the $x$ and $y$ axes is one such, represented by $z=0$. The plane $x+y+z=0$ is another, tilted and going through the origin. Each side of such a plane is a half space. The same happens in higher dimensions. If the coordinates are $x_1, x_2, \ldots ,x_n$, there is a hyperplane $x_1=5$ which divides the space into two half spaces: one with all points that satisfy $x_1 \gt 5$ and one with the ones that have $x_1 \lt 5$. Similarly, there is another with $x_1+4x_2+3x_3 = 9$ which is inclined in those axes.

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Usually, half-plane is just another name to a half-space in $\mathbb{R}^{2}$. The general definition is half-spaces requires some understanding of linear algebra, and the wikipedia article covers it pretty well.

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Mathworld article on Half-space relates half-space and hyperplane clearly as(quoting from the article):

A half-space is that portion of an n-dimensional space obtained by removing that part lying on one side of an (n-1)-dimensional hyperplane.

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