Piecewise linear

From Academic Kids


In mathematics, a piecewise linear function

<math>f: \Omega \to V<math>,

where V is a vector space and <math>\Omega<math> is a subset of a vector space, is any function with the property that <math>\Omega<math> can be decomposed into finitely many convex polytopes, such that f is equal to a linear function on each of these polytopes.

A special case is when f is a real-valued function on an interval <math>[x_1,x_2]<math>. Then f is piecewise linear if and only if <math>[x_1,x_2]<math> can be partitioned into finitely many sub-intervals, such that on each such sub-interval I, f is equal to a linear function

f(x) = aI</sup>x + bI.

The absolute value function <math>f(x) = |x|<math> is a good example of a piecewise linear function. Other examples include the square wave, the sawtooth function, and the floor function.

Important sub-classes of piecewise linear functions include the continuous piecewise linear functions and the convex piecewise linear functions.

PL manifolds

The idea of a piecewise linear (PL) structure on a topological manifold M is used in geometric topology. Smooth manifolds have PL structures, but not conversely, in general. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. A more slick definition is to use a sheaf, locally isomorphic to the sheaf of piecewise linear functions on Euclidean space.

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