Geometric Mean

The geometric mean is a type of power mean. For a collection $\{a_1, a_2, \ldots, a_n\}$ of positive real numbers, their geometric mean is defined to be

$$\text{GM}(a_1, \ldots, a_n) = \sqrt[n]{a_1 a_2 \ldots a_n}.$$

For instance, the geometric mean of $4$ and $9$ is

$$\text{GM}(4,9) = \sqrt{4\cdot 9} = \sqrt{36} = 6$$

and the geometric mean of $8$, $27$, and $64$ is

$$\text{GM}(8, 27, 64) = \sqrt[3]{8\cdot 27\cdot 64} = 2\cdot 3 \cdot 4 = 24.$$

The terminology geometric mean comes from the fact that this quantity has a simple geometric interpretation. Consider the below picture, where $AH = a$ and $HC = b$.

The segment $BH$ has length $\sqrt{ab}$, which is the geometric mean of the lengths of $AH$ and $HC$.

From the Pythagorean theorem, one obtains the three equations

$$\begin{aligned} AH^2 + BH^2 &= AB^2\\ HC^2 + BH^2 &= BC^2\\ AB^2 + BC^2 &= AC^2. \end{aligned}$$

This implies

$$AH^2 + 2 BH^2 + HC^2 = AC^2$$

and substituting $AH = a$, $HC = b$ gives

$$a^2 + b^2 + 2BH^2 = (a+b)^2 = a^2 + b^2 + 2ab \implies BH = \sqrt{ab}.$$

Thus,

$$BH = \text{GM}(AH, HC).$$

In particular, consider drawing a semicircle with diameter $AB$ that encapsulates the given figure. Then the radius of the semicircle is $\frac{a+b}2$, and the radius must be greater than or equal to $BH$ (where equality occurs only when $B$ is directly above the semicircle's center). Hence,

$$\frac{a+b}{2} \ge \sqrt{ab}$$

for all positive real numbers $a$ and $b$. This is the two-variable case of the AM-GM inequality.

Observe that by taking logarithms, we have

$$\log \text{GM}(a_1, \cdots, a_n) = \log \sqrt[n]{a_1 a_2 \cdots a_n} = \frac{\log a_1 + \log a_2 + \cdots +\log a_n}n.$$

Hence, the logarithm of the geometric mean is the arithmetic mean of the logarithms.