考虑非传统成本的最佳运输措施
Optimal measure transportation with respect to non-traditional costs
摘要：
我们研究了两个措施之间的最优质量运输问题的一个非传统的费用函数，即成本$c$，可以达到值$+\infty$。我们定义了两个测度的$c$ -相容性和强- $c$ -相容性的概念，并且证明如果在度量之间存在有限成本计划则度量必须是$c$兼容的，并且如果另外两个测量是强$c$兼容的，则存在一个集中于$c$的一个$c$ -次梯度上的最优方案THABMM-类函数。这个功能就是所谓的计划的潜力。我们给出这个定理的两个证明，在略有不同的假设。在第一，我们利用的概念$c$ -路径有界性，强$c$ -兼容性意味着一个强连通性的结果与最佳映射的有向图。图的强连通性意味着支撑集的$c$ -循环单调性（从经典推理得出）保证了它的$c$ -路径有界性，这意味着，反过来，存在一个潜在的。在其中一个测度是离散的情况下，我们也给出了一个构造性的证明。这种方法采用了一个新的概念“霍尔多面体”，我们介绍和深入研究，其中我们适用于一个版本的布劳威尔的不动点定理，以证明在这种情况下的潜在的存在。
英文摘要：
We study optimal mass transport problems between two measures with respect to a non-traditional cost function, i.e. a cost $c$ which can attain the value $+\infty$. We define the notion of $c$-compatibility and strong-$c$-compatibility of two measures, and prove that if there is a finite-cost plan between the measures then the measures must be $c$-compatible, and if in addition the two measures are strongly $c$-compatible, then there is an optimal plan concentrated on a $c$-subgradient of a $c$-class function. This function is the so-called potential of the plan. We give two proofs of this theorem, under slightly different assumptions. In the first we utilize the notion of $c$-path-boundedness, showing that strong $c$-compatibility implies a strong connectivity result for a directed graph associated with an optimal map. Strong connectivity of the graph implies that the $c$-cyclic monotonicity of the support set (which follows from classical reasoning) guarantees its $c$-path-boundedness, implying, in turn, the existence of a potential. We also give a constructive proof, in the case when one of the measures is discrete. This approach adopts a new notion of `Hall polytopes', which we introduce and study in depth, to which we apply a version of Brouwer's fixed point theorem to prove the existence of a potential in this case.
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