There is an inequality:
$$\sqrt[n]{\prod_{i = 1}^{n}{(a_i+b_i)}} \geq \sqrt[n]{\prod_{i = 1}^{n}{a_i}} + \sqrt[n]{\prod_{i = 1}^{n}{b_i}}$$
which I even don't know its name. I'd like to have an ask of its name and usage.
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It is a special case of the Brunn-Minkowski Theorem. It isn't strange at all; it is a rather useful inequality in measure theory.
Umberto P.
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Thanks for your answer. I really need an explore on Wikipedia instead of Chinese math textbooks. – Chielo Newctle Nov 25 '14 at 05:22
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This is Mahler's inequality. We had it here in this question (see also the linked questions), and it was question A2 on the 2003 Putnam, so you can find some proofs in Kedlaya's archive.
As for its usage: It is sometimes used to prove the Brunn-Minkowski inequality (of which it is a special case, as Umberto P. noted). In that proof, one first proves this inequality, deduces Brunn-Minkowski for the case when both bodies are axis-aligned boxes, uses a clever induction to extend to finite unions of axis-aligned boxes, and then completes the proof by approximation. That proof, which is due to Ohmann, can be found in §8.2 of Gruber's Convex and Discrete Geometry.
