Editing Piecewise linear function (section)

==Specializations==

Important sub-classes of piecewise linear functions include the [[continuous function|continuous]] piecewise linear functions and the [[Convex function|convex]] piecewise linear functions.
In general, for every ''n''-dimensional continuous piecewise linear function <math>f : \mathbb{R}^n \to \mathbb{R}</math>, there is a 
: <math>\Pi \in \mathcal{P}(\mathcal{P}(\mathbb{R}^{n+1}))</math>

such that
: <math>f(\vec{x}) = \min_{\Sigma \in \Pi} \max_{(\vec{a}, b) \in \Sigma} \vec{a} \cdot \vec{x} + b.</math><ref>{{cite journal
 | last = Ovchinnikov | first = Sergei
 | arxiv = math/0009026
 | issue = 1
 | journal = Beiträge zur Algebra und Geometrie
 | mr = 1913786
 | pages = 297–302
 | title = Max-min representation of piecewise linear functions
 | volume = 43
 | year = 2002}}</ref>

If <math>f</math> is convex and continuous, then there is a 
: <math>\Sigma \in \mathcal{P}(\mathbb{R}^{n+1})</math>

such that
: <math>f(\vec{x}) = \max_{(\vec{a},b) \in \Sigma} \vec{a} \cdot \vec{x} + b.</math>