I have a problem proving when $d_p(A,x)=d_p(B,x)$, meaning x is equidistant set, x always intersects $(A+B)/2$ (midpoint).
$d_p(A,x)$ is the $p^{th}$ Minkowski norm.
I have plotted a figure for different $L_p$ norms below. Is there a way to prove that?
ADDITIONAL NOTE: I am new on this topic and also on this forum. I am not a mathematician but an engineer. Any suggestion on an academic paper or a text book related to this topic also appreciated.
If I interpret the question correctly, you are asking why the arithmetical mean $x=\frac{A+B}2$ has the same distance to $A$ and $B$.
This holds actually in any normed space: Just note that $$A-x=\frac{A-B}2=x-B\text,$$ which readily implies $\lVert A-x\rVert=\lVert x-B\rVert$.