 Solution

Every function $h$ with Lipschitz constant at most $1$ yields a feasible pair $(h,-h)$, and replacing $h$ by $-h$ reverses the objective. HenceConversely, the Lipschitz regularization of transport potentials in part iii dominates every feasible pair by $(h,-h)$ with $\Vert h{\Vert }_L\le 1$, so $m_L\le \gamma$. Together with the preceding parts this proves the Kantorovich–Rubinstein theorem here in all four requested forms:The seminorm convention is essential. If instead $\Vert h{\Vert }_L$ meant $\Vert h{\Vert }_{\infty }+\mathrm{Lip}(h)$, the equality would fail: on a two-point space at distance $2$, with point masses at the two different points, $W=2$ whereas the latter unit-ball supremum is $1$. Indeed, an oscillation $a$ needs supremum norm at least $a/2$ and Lipschitz constant $a/2$, so the sum norm bounds $a$ by $1$, attained by values $-1/2,1/2$. This distinguishes the Lipschitz seminorm used above from a full bounded-Lipschitz norm.