Applications of the `Ham Sandwich Theorem' to eigenvalues of the Laplacian

TL;DR

This paper uses Gromov's ham sandwich method to derive domain monotonicity, reverse domain monotonicity, and universal inequalities for Neumann eigenvalues of the Laplacian on convex domains.

Contribution

It introduces novel applications of the ham sandwich theorem to establish key inequalities and monotonicity properties of Laplacian eigenvalues in convex domains.

Findings

Domain monotonicity up to a constant factor

Reverse domain monotonicity up to a constant factor

Universal inequalities for Neumann eigenvalues

Abstract

We apply Gromov's ham sandwich method to get (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in a Euclidean space.