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I have a question as to what someone means when they speak about hyperplanes.

In "A combinatorial bound for linear programming and related problems", they mention hyperplanes without really defining them, and I would really like to know what they mean.

On Wikipedia, "hyperplane is a subspace of one dimension less than its ambient space", no mention of linearity.

Also, they define an "affine hyperplane", they mention something of the form:

[a_1 x_1 + a_2 x_2 + ... + a_n x_n = b]

However, Wolfram Mathworld seems to contradict this by simply defining a "hyperplane" with anequation similar to the one above.

Which is the definition that they use in the article I mentioned? (In the context of LP-type problems.)

user1582024
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1 Answers1

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In the real space of dimension $d$, $\mathbb{R}^d$, an affine hyperplane $\mathcal{H}_(a,b)$ is defined by some vector $(a_1,\ldots,a_d)$ and a constant $b$. It is $$ \mathcal{H}_(a,b) = \{ x \in \mathcal{R}^d : a_1x_1+ \ldots + a_dx_d = b\}.$$

A hyperplane would be any affine hyperplane with $b = 0$ (thus only defined by the vector $a$).

In linear programming any constraint is nothing but an affine hyperplane. So it is common to use the word hyperplane for affine hyperplanes. The context should clarify whether they speak of a hyperplane with respect to a vector and a constant (and thus speak of affine hyperplane) or with respect only to a vector (and thus speak of a hyperplane).

laurent
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