Multimarginal optimal transport (MOT) has gained increasing attention in recent years, notably due to its relevance in machine learning and statistics, where one seeks to jointly compare and align multiple probability distributions. This paper presents a unified and complete Kantorovich duality theory for MOT problem on general Polish product spaces with bounded continuous cost function. For marginal compact spaces, the duality identity is derived through a convex-analytic reformulation, that identifies the dual problem as a Fenchel-Rockafellar conjugate. We obtain dual attainment and show that optimal potentials may always be chosen in the class of $c$-conjugate families, thereby extending classical two-marginal conjugacy principle into a genuinely multimarginal setting. In non-compact setting, where direct compactness arguments are unavailable, we recover duality via a truncation-tightness procedure based on weak compactness of multimarginal transference plans and boundedness of the cost. We prove that the dual value is preserved under restriction to compact subsets and that admissible dual families can be regularized into uniformly bounded $c$-conjugate potentials. The argument relies on a refined use of $c$-splitting sets and their equivalence with multimarginal $c$-cyclical monotonicity. We then obtain dual attainment and exact primal-dual equality for MOT on arbitrary Polish spaces, together with a canonical representation of optimal dual potentials by $c$-conjugacy. These results provide a structural foundation for further developments in probabilistic and statistical analysis of MOT, including stability, differentiability, and asymptotic theory under marginal perturbations.

翻译：多边际最优运输（MOT）近年来受到越来越多的关注，特别是在机器学习和统计学领域，其重要性体现在需要联合比较和对齐多个概率分布。本文针对一般波兰乘积空间上具有有界连续成本函数的 MOT 问题，提出了一个统一且完整的 Kantorovich 对偶理论。对于边缘紧空间，通过对偶问题的凸分析重构，将其识别为 Fenchel-Rockafellar 共轭，从而推导出对偶恒等式。我们获得了对偶可达性，并证明最优势函数总可在 $c$-共轭族类中选择，从而将经典的双边际共轭原理真正推广到多边际情形。在非紧情形下，由于缺乏直接的紧性论证，我们基于多边际转移计划的弱紧性及成本的有界性，通过截断-紧致化过程恢复了对偶性。我们证明了在对偶值限制在紧子集上时保持不变，且容许的对偶族可正则化为一致有界的 $c$-共轭势函数。该论证依赖于对 $c$-分裂集及其与多边际 $c$-循环单调性等价性的精细运用。随后，我们在任意波兰空间上获得了 MOT 的对偶可达性及精确的原-对偶等式，并通过 $c$-共轭性给出了最优对偶势函数的典范表示。这些结果为 MOT 在概率与统计分析中的进一步发展（包括在边缘扰动下的稳定性、可微性及渐近理论）提供了结构基础。