Concept guide
Finite convex combination: checking a witness with exact weights, and separating negative weights, approximate weights and uniqueness
In ℝ² let v_1=(0,0), v_2=(6,0), v_3=(0,6). (1) Check that λ=(1/6,1/3,1/2) are convex weights and reconstruct w=(2,3); (2) show that zero weights are allowed and that two-point convex combinations are segment points; (3) prove that u=(4,4) is not a convex combination of v_1, v_2, v_3 (its only affine weights include −1/3), with a second proof by a linear-functional bound; (4) show that three weights of 0.333 do not qualify; (5) after adding v_4=(3,3), exhibit two different convex weight vectors for the same point.