Finite convex combination

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

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 — Vector addition and real scalar multiplication carry the combination.
• Ratio and proportion — Rational weights must satisfy the total-one condition.
• 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.