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

This means that it cannot be considered to contain or lead to any mathematically interesting information.

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 $\frac{1}{\epsilon} \delta(\frac{x-y}{\epsilon})$ .

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