# 16. Concavity (Convexity)

## GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Concavity (Convexity)

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III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

<< Previous page  TOC  INDEX  Next page >>

### Concavity (Convexity)

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

Definition. A functional is a mapping from a vector space to the real numbers .

Thus, for example, every norm is a functional.

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

Definition. The norm of a linear functional is defined on the normed linear space by

Definition. A functional defined on a concave subset of a vector space is said to be concave on 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 is strictly concave on if strict inequality holds above for . Finally, isuniformly concave on if there exists such that for all ,

We have

Definition. A local minimizer of a real-valued function is any such that in some neighborhood of .

Definition. A global minimizer of a real-valued function on a set is any such that for all .

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

Definition. The concave hull of a set in a metric space is the smallest concave set containing .

<< Previous page  TOC  INDEX  Next page >>