Convex function
Definition
Let be an interval and let be a function. The function is convex if for all and all .
Remarks
Geometrically, each chord of the graph lies on or above the graph.
For example, , , and are convex on . Linear functions are convex.
A twice differentiable function with is convex.
Mathlib defines convexity on a set as ConvexOn.
Where this leads
37 pages build on this one.
- A differentiable function is convex iff its derivative is increasing
- A convex function on an open interval is continuous
- 1 + x is at most exp x
- log x is at most x - 1
- Strictly convex function
- Slopes of a convex function are increasing
- The exponential function is convex
- Concave function
- Jensen's inequality
- Inflection point
- A function with nonnegative second derivative is convex
- The logarithm is strictly concave
- Doob's upcrossing inequality
- Hermite–Hadamard inequality
- Doob's martingale inequality
- Consistency of the maximum likelihood estimator
- Log-modulus of a holomorphic function is subharmonic
- Asymptotic normality of the maximum likelihood estimator
- Subderivative
- Log-convexity of the gamma function
- Huber loss
- Wald's equation
- Optional stopping theorem
- Convex functions of subharmonic functions
- Logarithmically convex function
- Submartingale
- Karamata's inequality
- Jensen's inequality for integrals
- Doob's martingale convergence theorems
- Bohr–Mollerup theorem
- Expectation–maximization algorithm
- Doob–Meyer decomposition theorem
- Hadamard three-circle theorem for subharmonic functions
- Jensen inequality as a functional inequality
- Skorokhod's embedding theorem
- Generalized mean inequality
- Doob decomposition theorem
Further reading
About this page
| Tag | 0VV0HSW1ZS |
|---|---|
| URI | m2://0VV0HSW1ZS |
| Kind | definition |
| Writing | W1 stated |
| MSC | 26A51 |
| Defines | Convex function |
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What this page needs for W2
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What links here
Used by
- A differentiable function is convex iff its derivative is increasing
- Strictly convex function
- Slopes of a convex function are increasing
- Concave function
- Jensen's inequality
- Inflection point
- Hermite–Hadamard inequality
- Subderivative
- Huber loss
- Logarithmically convex function
- Karamata's inequality
- Jensen's inequality for integrals
- Bohr–Mollerup theorem