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 (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...

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 the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...

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 place of the minimum, and the

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

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

piecewise In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be ...

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 sequences, and these envelopes can be computed efficiently by a divide-and-conquer algorithm 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 distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...

s or quasiconvex functions, 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 functions preserve the property of being Lipschitz. However, the lower and upper envelope operations do not necessarily preserve the property of being a

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

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