Given three set $A=(a_1, a_2), B=(b_1, b_2, b_3), C=(c_1, c_2, c_3)$. where $a_i, b_i, c_i \in \mathbb{R} $.
I want to find a binary partition for example $L=A\cup B , R= C $, such that the sum of their absolute deviation is minimized.
E.g. $\textbf{argmin}_{L,R} E(L) + E(R) = \sum_{i \in L}|i - median(L)| + \sum_{j \in R}|j - median(R)|$
Instead of finding the partition, I want to prove or disprove, if there always exists a function $f : \{A, B, C\} \to \mathbb{R}$.
Such that $f(A),f(B), f(C)$ would have the sample partition as $A, B, C$ with objective function minimized.
Namely, If I know the optimal partition in the original problem is $L = A\cup B, R = C$.
Then $L' = \{f(A), f(B)\}, R' = \{f(C)\}$. such that $\sum_{i \in L'}|i - median(L')| + \sum_{j \in R'}|j - median(R')|$ is minizimed.