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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Solution of the Monge problem II: Local approach
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In the previous chapter, we tried to establish the almost sure singlevaluedness of the
c
-subdifferential by an argument involving “global” topological properties, such as connectedness. Since this strategy worked out only in certain particular cases, we shall now explore a different method, based on
local
properties of
c
-convex functions. The idea is that the global question
“Is the c-subdifferential of ψ at x single-valued or not?”
might be much more subtle to attack than the local question
“Is the function ψ differentiable at x or not?”
For a large class of cost functions, these questions are in fact equivalent; but these different formulations suggest different strategies. So in this chapter, the emphasis will be on tangent vectors and gradients, rather than points in the
c
-subdifferential.