# Finite convex combination

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

A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one.

mko-finite-convex-combination · definition · v0.10.0



## Mathematical statement

A finite convex combination of vectors v_1,…,v_n is sum_i lambda_i*v_i, where n≥1, each lambda_i≥0 and sum_i lambda_i=1. Zero weights are allowed. Negative weights or a total only approximately one do not qualify as convex weights; the expression may instead be a general affine or linear combination.





## Representative formula (exact TeX)

C(V)





## Assumptions


- Exercises use integer coordinate vectors and exact rational weights, not floating-point approximations.
- A convex-combination witness establishes the given weighted point, not coordinate uniqueness without an additional affine-independence condition.





## Symbol scope


- C — convex-combination record
- lambda — exact weights





## Intuition, definition, and misconceptions

Qualification separately checks nonnegativity and total one.

A two-point combination uses lambda and 1−lambda to give a point on the segment.

A multi-point witness is audited by exact reconstruction of its target.

Do not allow negative weights, replace an inexact near-one total by exact 1 or infer unique coordinates merely from a convex combination.





## Adopted derivation guide

[Finite convex combination: checking a witness with exact weights, and separating negative weights, approximate weights and uniqueness](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-convex-combination-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — Vector addition and real scalar multiplication carry the combination.
- [Ratio and proportion](https://ocme.evemisslab.com/en/mko/mko-ratio-proportion/) — Rational weights must satisfy the total-one condition.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Weighting and reduction use exact arithmetic.





## Verification boundary

Only finite nonempty vector lists and exact rational weights are handled.

not_formalized

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





## Next obligation

Add convex sets, convex hulls and affine independence.





## 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/finite-convex-combination.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-convex-combination/index.md)