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Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer.

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Quantitative is typically used as opposed to qualitative. In the geometric context referred to by the OP, the distinction is explained as follows:

Quantitative geometry and topology refines the qualitative, discrete questions of algebraic and geometric topology into continuous ones. For example, we may see a loop in a space which is homotopically trivial and ask how difficult it is to trivialize. Depending on what we mean by "difficult", we might obtain different notions of isoperimetry; one common choice is the area of a filling disk, which leads to the definition of the Dehn function of a group.

A priori, such notions usually depend on the choice of a metric on the space; one can then analyze their dependence on the metric (quantitative geometry) or show results, for example asymptotic ones, which are independent of the choice (quantitative topology).

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  • $\begingroup$ Thanks for answering this. I've always been confused; though it seems still like a bit of a slang $\endgroup$
    – ABIM
    Oct 4, 2019 at 11:51
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    $\begingroup$ More generally it’s proving not just an inequality but an estimate of the difference in terms of interesting quantities related to the inequality. $\endgroup$
    – Deane Yang
    Oct 4, 2019 at 13:11

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