# Convex function

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A function on a convex domain is convex when its value at every two-point convex combination is at most the same weighted combination of endpoint values.

mko-convex-function · definition · v0.10.0



## Mathematical statement

On a convex domain C, f is convex if f(lambda*x+(1−lambda)*y)≤lambda*f(x)+(1−lambda)*f(y) for every x,y in C and every lambda in [0,1]. This slice uses complete algebraic certificates for intervals and univariate affine or quadratic functions.





## Representative formula (exact TeX)

J(f)





## Assumptions


- The domain is an explicitly declared nonempty real interval; quadratic coefficients and Jensen instances are exact.
- One Jensen violation suffices to refute convexity, but one passing instance cannot prove it. Positive certificates must cover every x, y and lambda.





## Symbol scope


- J — Jensen-convexity record
- f — real-valued function





## Intuition, definition, and misconceptions

An affine function gives equality for every Jensen combination.

The Jensen gap for a quadratic simplifies to a*lambda*(1−lambda)*(x−y)².

Finite instances supply counterexamples or illustrations, not replacements for universal certificates.

Do not use the usual convex-function qualification without a convex domain or infer global convexity from a few passing Jensen samples.





## Adopted derivation guide

[Convex function: a complete certificate via the Jensen gap of a quadratic, and a single Jensen counterexample](https://ocme.evemisslab.com/en/derivations/derivation-mko-convex-function-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Convex set](https://ocme.evemisslab.com/en/mko/mko-convex-set/) — A convex-function claim first requires a convex domain.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The function must be total on its domain.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Jensen conditions compare values by inequalities.





## Verification boundary

Only affine/quadratic algebraic certificates on one-variable intervals are checked.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add strict convexity, subgradients and higher-dimensional convex functions.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/convex-function.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-convex-function/index.md)