Convex function

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Convex set — A convex-function claim first requires a convex domain.
• Function mapping — The function must be total on its domain.
• Usual order on the real numbers — 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.