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- W1574288003 abstract "Monge’s problem refers to the classical problem of optimally transporting mass: given Borel probability measures <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu Superscript plus Baseline not-equals mu Superscript minus> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:msup> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mu ^+ ne mu ^-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, find the measure-preserving map <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s colon upper M long right-arrow upper M> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=false>⟶<!-- ⟶ --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>s:M longrightarrow M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> — and assuming absolute continuity of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu Superscript plus> <mml:semantics> <mml:msup> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>mu ^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> — an optimal map will be shown to exist. Aspects of its uniqueness are also established." @default.
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- W1574288003 date "2001-12-04" @default.
- W1574288003 modified "2023-09-25" @default.
- W1574288003 title "Monge’s transport problem on a Riemannian manifold" @default.
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- W1574288003 doi "https://doi.org/10.1090/s0002-9947-01-02930-0" @default.
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