Odd-distance and right-equidistant sets in the maximum and Manhattan metrics

TL;DR

This paper determines the maximum sizes of right-equidistant sequences and sets with pairwise odd distances in the max metric space, providing exact bounds and partial results for the Manhattan metric.

Contribution

It establishes exact maximum sizes for right-equidistant sequences and odd-distance sets in max metric space, advancing extremal geometry understanding.

Findings

Maximum size of right-equidistant sequences in $ ext{max metric}$ is $2^{n+1}-1$.

Maximum size of pairwise odd-distance sets in $ ext{max metric}$ is $2^n$.

Partial results are obtained for the Manhattan metric space.

Abstract

We solve two related extremal-geometric questions in the $n-$dimensional space $\mathbb{R}^n_{\infty}$ equipped with the maximum metric. First, we prove that the maximum size of a right-equidistant sequence of points in $\mathbb{R}^n_{\infty}$ equals $2^{n+1}-1$. A sequence is right-equidistant if each of the points is at the same distance from all the succeeding points. Second, we prove that the maximum number of points in $\mathbb{R}^n_{\infty}$ with pairwise odd distances equals $2^n$. We also obtain partial results for both questions in the $n-$dimensional space $\mathbb{R}^n_1$ with the Manhattan distance.