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In Inequality of arithmetic and geometric means, it says: "Intuitively this corresponds to the fact that the exponential function, which converts addition to multiplication, is strictly convex ...".

How is this meant?

Basti
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    Which part/word do you fail to understand in this statement? – Did Jun 21 '13 at 22:00
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    A convex function satisfies $ f\left(\frac{x + y}{2}\right) \le \frac{f(x) + f(y)}{2} $. That's the important part of the exponential function being convex. Then replace $ f $ with $ exp $ and use a bit of arithmetic to conclude. – Jon Claus Jun 21 '13 at 22:02
  • How does the exponential function "convert" addition to multiplication? – Basti Jun 21 '13 at 22:03

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Let $\exp$ denote the exponential function. Then $$\exp(a+b)=\exp(a)\cdot\exp(b).$$ In fancier language, $\exp$ is a group homomorphism from the real numbers $\mathbb{R}$ under addition, to the positive real numbers $\mathbb{R}_{>0}$ under multiplication.

Zev Chonoles
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