lacunary - Mathnotes

Convexity

Definition: Convex Function (also: convex) \@{convex-function}

Let E⊆Rk be a convex set. A function φ:E→R is convex if

φ(λx+(1−λ)y)≤λφ(x)+(1−λ)φ(y)

whenever x,y∈E and 0<λ<1.

In geometric terms, this means a @chord joining any two points on the graph of φ lies on or above the graph between them.

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Definition: Strictly Convex Function \@{strictly-convex-function}

Starting with convex function, φ is strictly convex if the inequality is strict whenever x≠y.

Definition: Concave Function (also: concave) \@{concave-function}

A function φ is said to be concave if −φ is a convex function.

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Theorem: Second Derivative Test for Convexity \@{second-derivative-test-for-convexity}

Let φ be twice differentiable on an open interval (a,b). Then φ is convex on (a,b) if and only if φ″(x)≥0 for all x∈(a,b), and concave if and only if φ″(x)≤0 for all x∈(a,b).

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Definition: Convex Combination \@{convex-combination}

A convex combination is a linear combination of points where all @coefficients are non-negative and sum to 1.

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Note \@{note-6}

Every convex combination of two points lies on the @line-segment between the points.