In this paper, we establish a Kantorovich duality for unbalanced optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also get another proof of Kantorovich–Rubinstein theorem for generalized Wasserstein distance $\widetilde {W}_1^{a,b}$ proved before by Piccoli and Rossi. Then we apply our duality formula to study generalized Wasserstein barycenters. We show the existence of these barycenters for measures with compact supports. Finally, we prove the consistency of our barycenters.

在本文中，我们为不平衡的最优总变差传输问题建立了 Kantorovich 对偶。因此，我们恢复了 Caffarelli 和 McCann 建立的部分最优传输的对偶公式版本；我们还得到了 Kantorovich-Rubinstein 定理的另一个证明，用于广义 Wasserstein 距离$\widetilde {W}_1^{a,b}$之前由 Piccoli 和 Rossi 证明。然后我们应用我们的对偶公式来研究广义 Wasserstein 重心。我们展示了这些重心的存在，用于具有紧凑支撑的措施。最后，我们证明了重心的一致性。