Polyhedral approximation of a convex function is shown to be controlled by the quantization error of the Monge-Ampère measure of its Legendre-Fenchel dual, yielding a k-center pruning algorithm.
The Monge-Amp\`ere equation and its link to optimal transportation
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abstract
We survey the (old and new) regularity theory for the Monge-Amp\`ere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Amp\`ere type equations arising in that context.
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Duality between polyhedral approximation of value functions and optimal quantization of measures
Polyhedral approximation of a convex function is shown to be controlled by the quantization error of the Monge-Ampère measure of its Legendre-Fenchel dual, yielding a k-center pruning algorithm.