Title: The infimal convolution structure of the Hellinger-Kantorovich distance Show Chinese title

Title: Hellinger-Kantorovich距离的最小卷积结构

Abstract: We show that the Hellinger-Kantorovich distance can be expressed as the metric infimal convolution of the Hellinger and the Wasserstein distances, as conjectured by Liero, Mielke, and Savar\'e. To prove it, we study with the tools of Unbalanced Optimal Transport the so called Marginal Entropy-Transport problem that arises as a single minimization step in the definition of infimal convolution. Careful estimates and results when the number of minimization steps diverges are also provided, both in the specific case of the Hellinger-Kantorovich setting and in the general one of abstract distances. Show Chinese abstract

Abstract: 我们证明了Hellinger-Kantorovich距离可以表示为Hellinger距离和Wasserstein距离的度量下卷积，这验证了Liero、Mielke和Savaré的猜想。 为了证明这一点，我们利用不平衡最优传输的工具研究了所谓的边缘熵-运输问题，该问题作为下卷积定义中的单一最小化步骤出现。 还提供了当最小化步骤数量发散时的详细估计和结果，既包括Hellinger-Kantorovich设置的具体情况，也包括抽象距离的一般情况。