Concept guide
Convex set: complete convexity certificates for a halfspace and an axis-aligned box, and a midpoint counterexample for a finite point set
In the real vector space ℝ²: (1) prove that the closed halfspace H={(x,y) ∣ 2x−3y≤6} is convex; (2) prove that the nonempty axis-aligned box B=[−1,2]×[0,3] is convex; both proofs must hold for all p, q and all λ∈[0,1]. (3) Refute the convexity of the finite point set F={(0,0),(2,0),(0,2)} with one exact midpoint witness, and explain why checking finitely many λ, or failing to find a counterexample, cannot prove convexity.