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!