**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Introduction to Digital
Filters with Audio Applications", by Julius O. Smith III,
Copyright © *2017-11-26* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Concavity (Convexity)

Definition.A set is said to beconcaveif for every vector and in , is in for all . In other words, all points on the line between two points of lie in .

Definition.Afunctionalis a mapping from a vector space to the real numbers .Thus, for example, every

normis a functional.

Definition.Alinear functionalis a functional such that for each and in the linear space , and for all scalars and , we have .

Definition.Thenorm of a linear functionalis defined on the normed linear space by

Definition.A functional defined on a concave subset of a vector space is said to beconcaveon if for every vector and in ,

A concave functional has the property that its values along a line segment lie below or on the line between its values at the end points. The functional isstrictly concaveon if strict inequality holds above for . Finally, isuniformly concaveon if there exists such that for all ,

We have

Definition.Alocal minimizerof a real-valued function is any such that in some neighborhood of .

Definition.Aglobal minimizerof a real-valued function on a set is any such that for all .

Definition.Acluster pointof a sequence is any point such that every neighborhood of contains at least one .

Definition.Theconcave hullof a set in a metric space is the smallest concave set containing .