Talk:Monge Problem

Introduction

• Perhaps replace the first sentence by something that would give the intuitive idea of the Monge problem to a non-mathematician. You can point out that the key difference between the Monge problem and the Kantorovich problem is that Monge doesn't allow mass to be split
• Change the phrase ``Kantorovich Problem`` to be a link to the corresponding page on the OT wiki

Background and Statement

• change the phrase ``Gaspard Monge`` to a link to his wikipedia page
• type-o ``the best way to move piles of dirt from its...``
• ``without separating any individual pile`` is a bit confusing -- perhaps somethign like ``all dirt that starts at the same initial location must be kept together and sent to the same final location``?
• ``a solution to the Monge Problem is a TRANSPORT map that specifies how to rearrange the dirt with minimal cost
• ``let X and Y be metric spaces``
• mu and nu must be probability measures
• ``called a transport map FROM MU TO NU``
• ``infimum is taken over all transport maps SENDING MU TO NU``

New section: Solutions

• add a new section explaining under what conditions solutions to the Monge problem exist/are unique

Issues with Formulation

• Replace the first part of the first sentence with``It took many years to prove existence of a solution to the Monge problem due to the constraint of only considering transport maps.``
• ``existence or uniqueness of AN OPTIMAL TRANSPORT MAP

Examples

• This section is empty. Please either add an example or remove this section.

Relation with Kantorovich

• ``the Kantorovich problem is framed in terms of transport plans, which allow mass starting at the same initial location to be split and sent to various final locations.``

References Section

• There is a broken reference to Monge's original paper