OAI.SymmetricPolar.symmetric_polar_main

The theorem states that for every dimension n ≥ 2 and every symmetric convex body K in n-dimensional Euclidean space (a compact convex set with nonempty interior such that x ∈ K if and only if −x ∈ K), the Gromov width of the polar product of K equals 4, and moreover that for every real c with 0 < c < 4 the open ball of capacity c admits a symplectic embedding into that polar product. Here the polar product is the Cartesian product, in phase space ℝⁿ × ℝⁿ, of the interior of K with the interior of its polar, where the polar of K is the set of p with ⟨q,p⟩ ≤ 1 for all q ∈ K. The capacity ball of capacity c is the set of phase points (x,y) with π(‖x‖² + ‖y‖²) < c. A symplectic embedding of U into V is a map e of phase space that is smooth (C^∞) on U, restricts to a topological embedding of U, sends U into V, and whose derivative at each point of U preserves the form ω₀(v,w) = ⟨v₁,w₂⟩ − ⟨w₁,v₂⟩. The Gromov width of a set is the supremum, as an extended nonnegative real, of those c > 0 for which the capacity ball of capacity c embeds symplectically into it.