Convexity

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1 · Prerequisites

Abstract Algebra

Linear Algebra

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval

Lemma

For a convex function and x<y<z, the three secant slopes satisfy s(x,y)≤ s(x,z)≤ s(y,z)

Theorem

A convex real function is Lipschitz on every closed bounded subinterval of the interior of its domain, hence continuous throughout the interior

Definition

The left and right derivatives of a real function as one-sided limits of its difference quotient

Theorem

A convex function on an open interval has finite left and right derivatives everywhere, with f'_-(u)≤ f'_+(u)≤ (f(v)-f(u))/(v-u)≤ f'_-(v)≤ f'_+(v) for u<v

Theorem

A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there

Remark

Convexity conventions, endpoint scope, dyadic approximation, and the exact use of the Axiom of Choice in the midpoint-convex counterexample

Sources

Convexity

3 · Items

Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval

For a convex function and x<y<z, the three secant slopes satisfy s(x,y)≤ s(x,z)≤ s(y,z)

A convex real function is Lipschitz on every closed bounded subinterval of the interior of its domain, hence continuous throughout the interior

The left and right derivatives of a real function as one-sided limits of its difference quotient

A convex function on an open interval has finite left and right derivatives everywhere, with f'_-(u)≤ f'_+(u)≤ (f(v)-f(u))/(v-u)≤ f'_-(v)≤ f'_+(v) for u<v

A convex function on an open interval is differentiable except at at most countably many points

A supporting line of slope m for a real function at an interior point

Every slope between the left and right derivatives of a convex function gives a supporting line

A differentiable function on an open interval is convex if and only if its derivative is nondecreasing

A differentiable convex function on an open interval has a continuous derivative

A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative

Midpoint convexity gives the convexity inequality at every dyadic weight k/2ⁿ

A continuous midpoint-convex function on an interval is convex

Finite Jensen inequality for a convex function and nonnegative weights summing to one

Every local minimum of a convex function on an interval is a global minimum

A strictly convex function has at most one global minimizer

An inflection point as a point of continuity where convexity changes to concavity or conversely

A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there

Convexity conventions, endpoint scope, dyadic approximation, and the exact use of the Axiom of Choice in the midpoint-convex counterexample