A symplectic capacity is a measure of the size of open subsets of phase space which is invariant under the action of symplectomorphisms, namely the symmetries of classical mechanics. The capacity of convex bodies of fixed volume is bounded from above by a universal constant and Viterbo conjectured that the bodies maximizing the capacity are exactly those symplectomorphic to a euclidean ball. In this talk, we will see that generalizing Viterbo's conjecture to paths of convex bodies yields a remarkable bi-invariant Lorentz-Finsler structure on the group of homogeneous symplectomorphisms. Already the restriction of this structure to the group of linear symplectomorphisms leads to many interesting open questions which we will outline towards the end of the talk. This is joint work in progress with Alberto Abbondandolo and Leonid Polterovich.

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