Convex function

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

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Definition

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

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

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Equivalent statements

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