Characterizing Distances Between Points in the Level Sets of a Class of Continuous Functions on a Closed Interval

TL;DR

This paper studies the distances between points in the level sets of continuous functions on a closed interval, revealing divisibility properties and minimum distances, with extensions to higher dimensions.

Contribution

It characterizes the set of distances between points with equal function values, establishing divisibility conditions and minimum distance constraints, and explores higher-dimensional generalizations.

Findings

Distances must divide the interval length.

The set of distances includes at least one third of the interval.

Higher-dimensional generalizations are proposed.

Abstract

Given a continuous function $f:[a,b]\to\mathbb{R}$ such that $f(a)=f(b)$, we investigate the set of distances $|x-y|$ where $f(x)=f(y)$. In particular, we show that the only distances this set must contain are ones which evenly divide $[a,b]$. Additionally, we show that it must contain at least one third of the interval $[0,b-a]$. Lastly, we explore some higher dimensional generalizations.