Let $X = \{x_1, \ldots, x_n\}$ be a set of points in $\mathbb{R}$
Define $L_p(X) := \text{argmin}_{x \in \mathbb{R}} \sum\limits_{x_i \in X} d(x, x_i)^p$
and let $L_\infty(X) := \text{argmin}_{x \in \mathbb{R}} \max_{x_i \in X} d(x, x_i)$
Prove that if $L_1(X) < L_\infty(X)$, then for all $1 \leq q < p < \infty$, we have $L_1(X) \leq L_q(X) < L_p(X) \leq L_\infty(X)$
We would like to show that the optimal points for minimizing each $L_p$-norm in 1D move monotonically between the median (argmin for $L_1$) and the mid-range (argmin for $L_\infty$) as the parameter $p$ increases. In the form written above it isn't exactly the $L_p$ norm because taking the $p^{th}$ root doesn't change the problem.