Abstract
Symplectic capacities coinciding on convex sets in the standard symplectic vector space are extended to any subsets of symplectic manifolds. It is shown that, using embeddings of non-smooth convex sets and a product formula, calculations of some capacities become very simple. Moreover, it is proved that there exist such capacities which are distinct and that there are star-shaped domains diffeomorphic to the ball but not symplectomorphic to any convex set. 1. Preliminaries. For an introduction to symplectic capacities, non-smooth Hamiltonian systems and characteristic differential inclusions we refer to a previous talk given at the Banach Center in October 93 [K93]. The aim of this note is to show that some calculations of symplectic capacities can be simplified through embeddings of non-smooth convex sets . No approximations by families of Hamiltonian functions are needed. We show that definitions of capacities of convex sets in the symplectic model space ( R, ω ) suffice to define and to calculate in some cases symplectic capacities for subsets in any symplectic manifolds. Moreover, some applications of the product formula for convex sets derived in [K90] are given. To define the setting, let us consider the standard symplectic linear space V :=
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