# A Unified Kantorovich Duality for Multimarginal Optimal Transport

arXiv: https://arxiv.org/abs/2601.17171 · HTML: https://mlsanity.com/paper/2601.17171/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 39 references that are in the map.
- First posted on arXiv 23 Jan 2026.

## Builds on (most cited first, 8 of 8)

- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [AttGAN: Facial Attribute Editing by Only Changing What You Want](https://mlsanity.com/paper/1711.10678/) (2017)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [Multi-marginal optimal transport: theory and applications](https://mlsanity.com/paper/1406.0026/) (2014)
- [Multi-marginal Wasserstein GAN](https://mlsanity.com/paper/1911.00888/) (2019)
- [Unsupervised Multi-Domain Image Translation with Domain-Specific Encoders/Decoders](https://mlsanity.com/paper/1712.02050/) (2017)
- [c-cyclical monotonicity as a sufficient criterion for optimality in the multi-marginal Monge-Kantorovich problem](https://mlsanity.com/paper/1601.05608/) (2016)
- [Monge solutions and uniqueness in multi-marginal optimal transport via graph theory](https://mlsanity.com/paper/2104.09488/) (2021)

## Abstract (arXiv)

We study Kantorovich duality for multimarginal optimal transport (MOT) with bounded continuous cost functions. The main focus is not only the equality between the primal and dual values, but also the structure of optimal dual potentials. For compact metric spaces, we prove that the dual problem admits an optimizer in the class of mutually c-conjugate families. The proof combines a Fenchel--Rockafellar duality argument with equicontinuity estimates, a normalization of the dual potentials, and the Arzel`a--Ascoli theorem. We then consider the non-compact case, where the marginal spaces are Polish. We first recover the Kantorovich duality identity by a truncation and tightness argument, using the boundedness of the cost and Prokhorov compactness. For dual attainment, we rely on the geometry of optimal supports. Under a natural support-splitting condition, we obtain a bounded Borel measurable dual optimizer in the canonical class of mutually c-conjugate families. Thus, the paper identifies mutually c-conjugate potentials as natural representatives of optimal dual families in the multimarginal setting. These results provide a structural basis for further work on stability, empirical multimarginal transport, and statistical limit theory.