Optimal Transport in One Dimension

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In this article, we explore the optimal transport problem on the real line along with some examples.

Linear Cost Example

For this example, consider the cost function along with a given linear map . Moreover, if let be any transport plan, then by direct computation we see that

                                                                     

which suggests that this result only depends on the marginals of (wherein and are compactly supported probability measures). In fact, in such cases, every transport plan/map is optimal.

Distance Cost Example

Consider the cost function along with probability measures (on ) and . Then, for any we see that , which then immediately puts us back in the linear cost position, so any transport map/plan is also optimal for such costs.

Book Shifting Example

Consider the cost function along with and (where is the one-dimensional Lebesgue measure). A (monotone) transport plan that rearranges to look like is given by and its corresponding cost is

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Furthermore, notice that the piecewise map given by (for ) and (for ) satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_1 # \mu = \nu }