Level curves of $f(r) = \sum_{i=1}^na_id(r,r_i)$ with $r_i\in \mathbb{R}^n$

Question

Let $$f(r) = \sum_{i=1}^na_id(r,r_i)$$ Where each $r_i \in \mathbb{R}^n$ and each $a_i\in \mathbb{R}$ and $d:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ denotes the usual distance function. That is, $f$ is a linear function of the distance from $n$ points in $n$-dimensional space. What are the level surfaces (curves) of $f$? When $n=2$, I believe these are conic sections (please correct me if I'm wrong about this). Are there analogs for $n>2$? If this cannot be answered in full generality, (which I'm guessing it can't) what area of math deals with such questions?

Answer 1

Information about the two dimensional case.

Let $A$ and $B$ be points in the plane and suppose $a+b = 1$. Define $$ f(X) = a \cdot d(X,A) + b \cdot d(X,B). $$ Then the level curves of $f$ are convex ovals surrounding $A$ and $B$. They are described by fourth degree equations in the coordinates that result when you start by squaring both sides of $$ c - a \cdot d(X,A) = b \cdot d(X,B). $$ That will leave a square root on the left. Move all the other terms on the left to the right and square again to see the fourth degree expression. It's not pretty.

When $a = b = 1/2$ the level curves are ellipses with foci $A$ and $B$: the third and fourth order terms cancel.

When $a=1, b=0$ the level curves are circles centered at $A$.

When $c$ is very large $A$ and $B$ are close together relative to $c$. The level curve looks more like a circle.