Article revision ideas

Below, you can find a list of revision ideas for articles. If you choose to write about one of these ideas, remove it from the list below and email me.

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List of Articles

Monge Problem

• The definition of a transport map and the $t\#\mu =\nu$ notation should be highlighted, possibly in a separate section or definition environment.
• The equation expressing the Monge problem could be improved -- what is the infimum taken over? Why is the function F(T) defined within the infimum?
• There are errors in the formatting of the references.
• The "Issues with Formulation" section could be renamed "Challenges of Monge Problem". This section could be expended.
• The relation with the Kantorovich problem should be expanded. A citation to the wiki article no the Kantorovich article should be included. Explain precisely why the Kantorovich problem generalizes the Monge Problem.
• Add a section stating Brenier's theorem on solvability of the Monge problem and relation to Kantorovich's problem. Cite work by Gigli (see Santambrogio for the reference) on why Failed to parse (unknown function "\math"): {\displaystyle \mu<\math> absolutely continuous with respect to Lebesgue may be weakened to not giving mass on sets of d-1 dimensional Hausdorff measure. [[Kantorovich Problem]] * This page refers to the Monge problem too much. It should stand on its own. A separate section should be added that includes all the information about the relationship to the Monge problem. * I don't think the dual problem should be mentioned in the introduction. This should be mentioned later. Perhaps the ``Introduction'' section should be removed. I don't think the Shipper's problem needs to be mentioned here. * There are some grammatical errors that should be fixed. * Intuition for the meaning of the transport plan should be given -- <math>\pi(A \times B)} represents the amount of mass that is sent from A to B.
• The statement of the more problem could be improved. The formatting of this problem makes it a little hard to see that it is a minimization problem.
• The section on Kantorovich Duality should link to the wiki article on Kantorovich Duality.
• A section should be added explaining how the Kantorovich article can be solved by direct menthod of the Calculus of Variations.
• A section should be added stating the Knott-Smith optimality criterion for OT plans.

Optimal Transport in One Dimension

• Suggested changes TBD

Kantorovich Dual Problem (for general costs)

• This article should be merged with article on the Kantorovich Dual Problem for c= d^2
• Add a discussion on the dual problem for c=d.
• Further suggested changes TBD

Discrete Optimal Transport

• Revision can begin on February 26
• Suggested changes TBD

Fenchel-Rockafellar and Linear Programming

• Suggested changes TBD

Optimal Transport and Ricci curvature

• Revision can begin on February 26
• Add a new section to the article discussing the relationship between Ricci curvature and convexity of the entropy. You should link to the existing wiki article on Geodesics and generalized geodesics.

Martingale optimal transport and mathematical finance

• Introduce some background material to explain the notation of what it means to be the expectation with respect to a filtration.
• Give intuition behind item (ii) in teh definition of an equivalent martingale measure
• Provide more intuition/background behind Problem 1 and 2. For example in Problem 2, what does it mean to be a martingale under Q? Why is that a natural requirement in this context?

Isoperimetric inequality and OMT

• Add sections describing recent work on quantitative isoperimetric inequalities, [1], [2]

The Moreau-Yosida Regularization

• Revision can begin on February 26
• Suggested changes TBD

Wasserstein barycenters and applications in image processing

• Revision can begin on February 26
• Suggested changes TBD

Wasserstein metric

• Revision can begin on February 26
• Suggested changes TBD