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

Abstract

We solve two related extremal-geometric questions in the $n-$dimensional space ${\mathbb{R}}_{\infty }^n$ equipped with the maximum metric. First, we prove that the maximum size of a right-equidistant sequence of points in ${\mathbb{R}}_{\infty }^n$ 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 ${\mathbb{R}}_{\infty }^n$ with pairwise odd distances equals $2^n$. We also obtain partial results for both questions in the $n-$dimensional space ${\mathbb{R}}_1^n$ with the Manhattan distance.

10 pages. v2: a few modifications based on the reviews

References in corpus (1)

• A problem of Kusner on equilateral sets

Cited by in corpus (1)

• Max-norm Ramsey Theory