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

\[\begin{align} AH^2 + BH^2 &= AB^2\\ HC^2 + BH^2 &= BC^2\\ AB^2 + BC^2 &= AC^2. \end{align}\]

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.