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

2022/06/21 by Yuanming Luo, Luo, Yuanming, Henry Riely +1
Computer Science · Mathematics · #26A06 #Advanced Optimization Algorithms Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2206.10494

openalex publication_date 2022/06/21 · openalex created_date 2022/06/24 · openalex updated_date 2026/07/28

Abstract

Given a continuous function f:[a,b]→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.

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