# A Hölder-type inequality for the Hausdorff distance between Lagrangians

## TL;DR

This paper proves a new inequality related to distances between Lagrangians in symplectic geometry.

## Contribution

A Hölder-type inequality is established for the Hausdorff distance between Lagrangians under specific metrics.

## Key findings

- A Hölder-type inequality is proven for the Hausdorff distance between Lagrangians.
- The inequality applies to the Lagrangian spectral distance and the Hofer–Chekanov distance.
- The result is derived using previously developed methods in symplectic geometry.

## Abstract

We prove a Hölder-type inequality (in the spirit of Joksimović and Seyfaddini in Int Math Res Not IMRN 8:6303–6324, 2024) for the Hausdorff distance between Lagrangians with respect to the Lagrangian spectral distance or the Hofer–Chekanov distance. This inequality is established via methods developed by the first author (Chassé in Int J Math 34(5):2350024, 2023; Chassé in Differ Geom Appl 94:Paper No. 102123, 22, 2024) to understand the symplectic geometry of certain collections of Lagrangians under metric constraints.