In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the

pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...

minimum 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 to

partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...

s by taking the minimum only among functions that have values at the point. The upper envelope or pointwise maximum is defined symmetrically. For an infinite set of functions, the same notions may be defined using the

infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...

in place of the minimum, and the

supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...

in place of the maximum. For continuous functions from a given class, the lower or upper envelope is a

piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...

function 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 using

Davenport–Schinzel sequence In combinatorics, a Davenport–Schinzel sequence is a sequence of symbols in which the number of times any two symbols may appear in alternation is limited. The maximum possible length of a Davenport–Schinzel sequence is bounded by the number of ...

s, and these envelopes can be computed efficiently by a

divide-and-conquer algorithm In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved dire ...

that computes and then merges the envelopes of subsets of the functions. For

convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...

s or

quasiconvex function In mathematics, a quasiconvex function is a real number, real-valued function (mathematics), function defined on an interval (mathematics), interval or on a convex set, convex subset of a real vector space such that the inverse image of any s ...

s, the upper envelope is again convex or quasiconvex. The lower envelope is not, but can be replaced by the lower convex envelope to obtain an operation analogous to the lower envelope that maintains convexity. The upper and lower envelopes of

Lipschitz function In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...

s preserve the property of being Lipschitz. However, the lower and upper envelope operations do not necessarily preserve the property of being a continuous function.

References

{{reflist, refs= {{citation , last1 = Boissonnat , first1 = Jean-Daniel , author1-link = Jean-Daniel Boissonnat , last2 = Yvinec , first2 = Mariette , author2-link = Mariette Yvinec , contribution = 15.3.2 Computing the lower envelope , contribution-url = https://books.google.com/books?id=Ax50ccq_kFAC&pg=PA358 , isbn = 9780521565295 , page = 358 , publisher = Cambridge University Press , title = Algorithmic Geometry , year = 1998 {{citation , last = Choquet , first = Gustave , author-link = Gustave Choquet , contribution = 3. Upper and lower envelopes of a family of functions , contribution-url = https://books.google.com/books?id=BmVwQYR6JqgC&pg=PA129 , isbn = 9780080873312 , pages = 129–131 , publisher = Academic Press , title = Topology , year = 1966 Functional analysis