The claim
Let a transport T have the distortion profile D(T) = (D₁(T), …, D_k(T)), each component a real number, smaller meaning less distortion. Following TERM-043, T is at least as faithful as T′ when Dⱼ(T) ≤ Dⱼ(T′) for every j. Say the two trade off when there are components i and j with Dᵢ(T) < Dᵢ(T′) and Dⱼ(T) > Dⱼ(T′). For weights w with every wⱼ > 0, write D_w = Σⱼ wⱼDⱼ.
- T and T′ are incomparable exactly when they trade off.
- If they trade off, some positive w gives D_w(T) < D_w(T′) and another gives D_w(T) > D_w(T′).
- If T is at least as faithful as T′ and strictly less distorted on some component, every positive w gives D_w(T) < D_w(T′).
Proof
Let a = D(T) − D(T′). (1) T is not at least as faithful as T′ exactly when some aⱼ > 0, and T′ is not at least as faithful as T exactly when some aᵢ < 0; both together are the trade-off. (2) Put weight M on a component with aᵢ < 0 and weight 1 on the rest: w·a ≤ M aᵢ + Σ_{l≠i} |a_l|, which is negative for M large. Putting M on a component with aⱼ > 0 instead makes w·a positive. (3) Every aⱼ ≤ 0 and some aⱼ < 0, so w·a < 0 when every wⱼ > 0.
What it does not say
- It does not say any pair of translations trades off. Whether the components move together for the tasks studied is QUESTION-013.
- It does not say how the components are measured (QUESTION-005), or that they are independent.
- It does not say no weighting is right. That fidelity is task-relative, and that the theory fixes no weighting, is CLAIM-128’s thesis, which is argued, not proved.