Convex function

See convex for other uses of the word (eg a convex set)

Stub grade: A*

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Demote once a reference is found and the definition is suitably abstracted

Definition

Let $S\in\mathcal{P}(\mathbb{R}^n)$ be an arbitrary subset of Euclidean $n$-space, $\mathbb{R}^n$, and let $f:S\rightarrow\mathbb{R}$ be a function. We say $f$ is a convex function if both of the following holdTemplate:RAFCIRAPM:

1. $S$ is a convex set itself, i.e. the line connecting any two points in $S$ is also entirely contained in $S$
   • In symbols: $\forall x,y\in S\forall t\in [0,1]\subset\mathbb{R}[x+t(y-x)\in S]$, and
2. The image of a point $t$-far along the line $[x,y]$ is $\le$ the point $t$-far along the line $f(x)$ to $f(y)$
   • In symbols: $\forall t\in [0,1]\subset\mathbb{R}[f(x+t(y-x))\le f(x)+t(f(y)-f(x))]$

Grade: C

This page requires some work to be carried out

Some aspect of this page is incomplete and work is required to finish it
The message provided is:

A picture would be great

Warning:That grade doesn't exist!

Equivalent statements

• "a function is convex if and only if its domain is convex and its epigraph are convex sets"Template:RAFCIRAPM