Observations on the Hofer distance between closed subsets

We prove the elementary but surprising fact that the Hofer distance between two closed subsets of a symplectic manifold can be expressed in terms of the restrictions of Hamiltonians to one of the subsets; this helps explain certain energy-capacity inequalities that appeared recently in [3] and [9]. We also build on [15] to obtain new vanishing results for the Hofer distance between subsets, applicable for instance to singular analytic subvarieties of Kähler manifolds.

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Accepted 7 July 2015

Published 23 May 2016