Definition:Convex Real Function/Definition 1

From ProofWiki
Jump to navigation Jump to search

Definition

Let $f$ be a real function which is defined on a real interval $I$.


$f$ is convex on $I$ if and only if:

$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$


ConvexFunction1.png


The geometric interpretation is that any point on the chord drawn on the graph of any convex function always lies on or above the graph.


Strictly Convex

$f$ is strictly convex on $I$ if and only if:

$\forall x, y \in I, x \ne y: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} < \alpha \map f x + \beta \map f y$


Also presented as

By setting $\alpha = \lambda$ and $\beta = 1 - \lambda$, this can also be written as:

$\forall x, y \in I, x \ne y: \forall \lambda \in \openint 0 1: \map f {\lambda x + \paren {1 - \lambda} y} \le t\lambda \map f x + \paren {1 - \lambda} \map f y$


Also known as

A convex function can also be referred to as:

a concave up function
a convex down function.


Also see


Sources


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
In particular: with respect to Also presented as
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Convex_Real_Function/Definition_1&oldid=663778"