# Favourite distances in 3-space

## TL;DR

This paper investigates the maximum sum of distances in a 3D point set with assigned radii, showing near all points lie on a circle with a specific symmetry, and provides an asymptotic formula for this extremal value.

## Contribution

It characterizes the structure of point sets that maximize the sum of distances with assigned radii in 3-space and derives an asymptotic formula for the maximum sum.

## Key findings

- Maximizers are mostly contained in a circle with a symmetry axis.
- The maximum sum of distances is asymptotically $n^2/4 + 5n/2 + O(1)$.
- A new construction matches the asymptotic bound.

## Abstract

Let $S$ be a set of $n$ points in Euclidean $3$-space. Assign to each $x\in S$ a distance $r(x)>0$, and let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erd\H{o}s and Pach (1988) introduced the extremal quantity $f_3(n)=\max\sum_{x\in S}e_r(x,S)$, where the maximum is taken over all $n$-point subsets $S$ of 3-space and all assignments $r\colon S\to(0,\infty)$ of distances. We show that if the pair $(S,r)$ maximises $f_3(n)$ and $n$ is sufficiently large, then, except for at most $2$ points, $S$ is contained in a circle $\mathcal{C}$ and the axis of symmetry $\mathcal{L}$ of $\mathcal{C}$, and $r(x)$ equals the distance from $x$ to $C$ for each $x\in S\cap\mathcal{L}$. This, together with a new construction, implies that $f_3(n)=n^2/4 + 5n/2 + O(1)$.