Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 348 / 3 / b

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- Solution b

Solution

Fix any admissible pair of Kantorovich potentials

(u, v)

and any transport plan

π \in Π (μ, ν)

. Its marginal distributions give

\int_{X} u d μ + \int_{Y} v d ν = \int_{X \times Y} (u (x) + v (y)) d π (x, y) .

The right side is well defined because

u (x)

and

v (y)

are Lebesgue integrable with respect to

π

. Integrating their pointwise feasibility inequality gives

\int_{X} u d μ + \int_{Y} v d ν \leq \int_{X \times Y} c (x, y) d π (x, y) .

Since this holds for every feasible pair and every transport plan,

(u, v) \in A_{c} sup (\int u d μ + \int v d ν) \leq π \in Π (μ, ν) in f K (π) .

This proves the required inequality directly from the transport constraints, without any convex optimization duality theorem. Whenever the extrema are attained, the supremum and infimum can respectively be written as a maximum and minimum. Even the Kantorovich duality theorem is unnecessary for this direction.

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