# Geometric mean, harmonic mean and loss functions

Consider a sequence $(x_i)_{i \in I}$ of real numbers indexed on a set $I$. The mode of the series is the minimizing argument for the $L_0$ loss

$$\text{mode}[ \; (x_i)_{i \in I} \; ] = \arg\min_{u \in \mathbb{R}} \; \; \sum_{i \in I} \; | x_i - u|^0$$

The median is the minimizing argument for the $L_1$ loss

$$\text{median}[ \; (x_i)_{i \in I} \; ] = \arg\min_{u \in \mathbb{R}} \; \; \sum_{i \in I} \; | x_i - u|^1$$

The arithmetic mean is the minimizing argument for the $L_2$ loss

$$\text{arithmetic mean}[ \; (x_i)_{i \in I} \; ] = \arg\min_{u \in \mathbb{R}} \; \; \sum_{i \in I} \; | x_i - u|^2$$

Can we find similar results for the harmonic mean or the geometric mean?

Thanks!

$$\arg \min_{u \in \mathbb{R}^+} \sum_{i \in I}|\log \frac{x_i}{u}|^2$$