L-Infinity Norm

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A L-Infinity Norm is a vector norm defined on the L-Infinity Space.


References

2015

[math]\displaystyle{ \left\| x \right\| _\infty = \max \left\{ |x_1|, |x_2|, \dotsc, |x_n| \right\} }[/math]

1999

[math]\displaystyle{ x=\begin{bmatrix}x_1 \\ x_2 \\ x_n \end{bmatrix} }[/math]
with complex entries by
[math]\displaystyle{ |x|_\infty=max_{i}|x_i| }[/math]

2011

[math]\displaystyle{ ||x||_\infty = sup(|x_1|, |x_2|,... |x_n|) }[/math]
when we search for neighbors. In this search, we are looking for all neighbors of a point at the center of a hypercube of fixed size in a vector space. Because we are concerned with finite-dimensional vector spaces in practice, we will use max() instead of sup() from now on.
Definition 1 A hypercube description region (HDR) is the set of points less than a fixed distance from a single point (called the center) using the L∞ norm. A weighted hypercube description region is an HDR that uses the positively weighted L∞ norm:
[math]\displaystyle{ ||x||_\infty = max(w_1|x_1|, w_2|x_2|,...w_n|x_n|) }[/math]
We will assume the term HDR refers to this more general case. Our use of weights implies that different points in a high-dimensional space can have different weights defining their hypercubes.
Definition 2 A composite hypercube description region (CHDR) is the set of points inside the union of zero or more hypercube description regions.