In mathematics, thelower envelopeorpointwise minimumof a finite set of functions is thepointwiseminimum of the functions, the function whose value at every point is the minimum of the values of the functions in the given set. The concept of a lower envelope can also be extended topartial functionsby taking the minimum only among functions that have values at the point. Theupper envelopeorpointwise maximumis defined symmetrically. For an infinite set of functions, the same notions may be defined using theinfimumin place of the minimum, and thesupremumin place of the maximum.[1]
For continuous functions from a given class, the lower or upper envelope is apiecewisefunction whose pieces are from the same class. For functions of a single real variable whose graphs have a bounded number of intersection points, the complexity of the lower or upper envelope can be bounded usingDavenport–Schinzel sequences,and these envelopes can be computed efficiently by adivide-and-conquer algorithmthat computes and then merges the envelopes of subsets of the functions.[2]
Forconvex functionsorquasiconvex functions,the upper envelope is again convex or quasiconvex. The lower envelope is not, but can be replaced by thelower convex envelopeto obtain an operation analogous to the lower envelope that maintains convexity. The upper and lower envelopes ofLipschitz functionspreserve the property of being Lipschitz. However, the lower and upper envelope operations do not necessarily preserve the property of being acontinuous function.[3]
References
edit- ^Choquet, Gustave(1966),"3. Upper and lower envelopes of a family of functions",Topology,Academic Press, pp. 129–131,ISBN9780080873312
- ^Boissonnat, Jean-Daniel;Yvinec, Mariette(1998),"15.3.2 Computing the lower envelope",Algorithmic Geometry,Cambridge University Press, p. 358,ISBN9780521565295
- ^Choquet (1966),p. 136.