Shadows of Symplectic Balls

The purpose is to give an overview on the squeezing and non-squeezing phenomena in symplectic geometry. Taking our lead to Gromov, we ask when the volume of a 2k-dimensional symplectic projection of a 2n-dimensional symplectic ball, i.e. the image of the unit ball via a symplectomorphism, is greater than or equal to pi^k/k!. We show that it holds for all k if the symplectomorphism is affine. In the non-affine case we prove the inequality for k=1 (the Gromov's Non-Squeezing Theorem) and for k=n (Liouville's Theorem). For 1<k<n we give a proof of a non-linear squeezing theorem.