A Hölder-type inequality for the Hausdorff distance between Lagrangians

We prove a Hölder-type inequality for the Hausdorff distance between Lagrangians with respect to the Lagrangian spectral distance or the Hofer-Chekanov distance in the spirit of Joksimović-Seyfaddini [arXiv:2207.11813]. This inequality is established via methods developped by the first author [arXiv:2204.02468, arXiv:2108.00555] in order to understand the symplectic geometry of certain collections of Lagrangians under metric constraints.