Definition:Hölder Mean/Zero Exponent

Definition
Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be positive real numbers.

Let $p$ be an extended real number.

Let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.

For $p = 0$, the Hölder mean is defined as:
 * $\map {M_0} {x_1, x_2, \ldots, x_n} = \paren {x_1 x_2 \cdots x_n}^{1 / n}$

which is the geometric mean of $x_1, x_2, \ldots, x_n$.

Also see

 * Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean, which justifies this definition