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To make a function less variable without changing it much, compare it with less variable functions

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Quick description

You are given a (bad) continuous real function f on a compact metric space (X,d); suppose you have a tool that applies well for Lipschitz functions, with good monotonicity properties; Then compare f with Lipschitz dominators: For t>0 let f_t(x) = \sup_y f(y) - t d(x,y). Then f_t\geq f is the minimum Lipschitz dominator for f with Lipschitz constant t.

General discussion

This is a lot like convolution with an approximate delta function; making the translations

  • \int \rightarrow \sup

  •  + \rightarrow \max

  • \times \rightarrow +

indicates that we've replaced the ring of real numbers with the tropical rig, for which  - \frac{1}{\epsilon} d(x,y) behaves a lot like the delta-function approximation  \frac{1}{\epsilon} \delta(\frac{x-y}{\epsilon}).

Comments

This looks nice, but do you

This looks nice, but do you plan to give some examples of problems where this trick is useful? It would be a big help.

useful!?

Oh yes... "tricks" wiki... yeah, it came up in a complex/harmonic analysis course I was taking, but the context... remains elusive.

Closely related is P

Closely related is Péron's expression solving the harmonic Dirichlet problem, as an infimum of sup-harmonic dominators.