Optimal transport of vector measures

Area of Science:

• Mathematics
• Optimal Transport Theory
• Measure Theory

Background:

• Optimal transport theory typically deals with scalar measures.
• A conjecture by Klartag proposed properties for vector measures with zero total mass.
• Understanding transport sets is crucial in various mathematical fields.

Purpose of the Study:

• To develop and study a theory of optimal transport for vector measures.
• To investigate Klartag's conjecture regarding vector measures with zero total mass.
• To generalize the Kantorovich-Rubinstein duality to the vector measures setting.

Main Methods:

• Development of a novel theory for optimal transport of vector measures.
• Construction of a specific counterexample to Klartag's conjecture.
• Generalization of the Kantorovich-Rubinstein duality for vector measures.

Main Results:

• A counterexample is provided, resolving Klartag's conjecture in the negative.
• The generalized Kantorovich-Rubinstein duality is established for vector measures.
• The conjecture is affirmed under the condition of absolutely continuous marginals.

Conclusions:

• Klartag's conjecture is disproven for general vector measures.
• The established duality provides new insights into optimal transport for vector measures.
• The findings offer a conditional affirmative answer to the conjecture, highlighting the importance of marginal properties.