<s>[INST] Can anything be said about the roots of the L4 center? Modes, Medians and Means: A Unifying Perspective defines the following centers based on the $L_p$ norms:
$$
\begin{aligned}
\text{mode of x} = \arg \min_s \sum_i \lvert x_i - s \rvert^0 \\
\text{median of x} = \arg \min_s \sum_i \lvert x_i - s \rvert^1 \\
\text{mean of x} = \arg \min_s \sum_i \lvert x_i - s \rvert^2 \\
\end{aligned}
$$
Where:
$$
\begin{aligned}
x = (x_1, x_2, \ldots, x_n) \ &:\ \text{a list of rational numbers} \\
\lvert x \rvert \ &:\ \text{number of elements in the list if $x$ is a list} \\
\end{aligned}
$$
I wanted to extend this to the L4 center:
$$
\begin{aligned}
\arg \min_s \sum_i \lvert x_i - s \rvert^4 = \text{L4 center} \\
\frac{d}{ds} \sum_i (x_i - s)^4 = 0 \\
\sum_i \left[ \frac{d}{ds}(x_i-s)^4 \right] = 0 \\
\sum_i \left[ -4(x_i-s)^3 \right] = 0 \\
\sum_i (x_i-s)^3 = 0 \\
\sum_i \left[ x_i^3 -3x_i^2s + 3x_is^2 - s^3 \right] = 0 \\
\sum_i x_i^3 + \sum_i -3x_i^2s + \sum_i 3x_is^2 + \sum_i -s^3 = 0 \\
\sum_i x_i^3 -3s \sum_i x_i^2 + 3s^2 \sum_i x_i - s^3 \lvert x \rvert = 0 \\
\end{aligned}
$$
Which is a cubic with the following discriminant:
$$
\begin{aligned}
&+ 162 \lvert x \rvert \sum_i x_i \sum_i x_i^2 \sum_i x_i^3 \\
&- 108 \left(\sum_i x_i\right)^3 \sum_i x_i^3 \\
&+ 18 \left(\sum_i x_i\right)^2 \left(\sum_i x_i^2\right)^2 \\
&- 108 \lvert x \rvert \left(\sum_i x_i^2\right)^3 \\
&- 27 \left(\lvert x \rvert\right)^2 \left(\sum_i x_i^3\right)^2
\end{aligned}
$$
I was wondering if anything could be said about the roots of the L4 center given any list $x$ of rationals? For example, how many real roots, because intuitively I expect only one?
 [/INST] The function $\sum_i(x_i-s)^3$ is strictly increasing in $s$, from $-\infty$ to $\infty$. It is also continuous, so there is a unique real root.
</s>