# Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues

## TL;DR

This paper investigates the optimal way to partition an interval to minimize the maximum of certain set-functions, with applications to Sturm-Liouville eigenvalues, proving existence, uniqueness, and asymptotic behavior of solutions.

## Contribution

It introduces a general framework for optimal interval partitioning under broad assumptions and connects it to classical Sturm-Liouville spectral results.

## Key findings

- Existence and uniqueness of optimal partitions
- Asymptotic distribution of partitions as n increases
- Recovery of classical Sturm-Liouville eigenvalue asymptotics

## Abstract

We study the optimal partitioning of a (possibly unbounded) interval of the real line into $n$ subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as $n$ tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function.