**Reference:**

**Geometric Inequalities, Vol 12, Mathematical Olympiad Series, Gangsong Leng, translated by Yongming Liu, East China Normal University Press, World Scientific.**

“God is always doing geometry”, said Plato. But, the deep investigation and extensive attention to geometric inequalities as an independent field is a matter of modern times.

Many geometric inequalities are not only typical examples of mathematical beauty but also tools for applications as well. The well known Brunn-Minkowski’s inequality is such an example. “It is like a large octopus, whose tentacles stretches out into almost every field of mathematics. It has not only relation with advanced mathematics such as the Hodge index theorem in algebraic geometry, but also plays an important role in applied subjects such as stereology, statistical mechanics and information theory.”

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