  Consider the image of a 2n-dimensional unit ball by an open symplectic
embedding into the standard symplectic vector space of dimension 2n. Its
2k-dimensional shadow is its orthogonal projection into a complex subspace of
real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is
at least the volume of the unit 2k-dimensional ball? This statement is
trivially true when k = n, and when k = 1 it is a reformulation of Gromov's
non-squeezing theorem. Therefore, this question can be considered as a
middle-dimensional generalization of the non-squeezing theorem. We investigate
the validity of this statement in the linear, nonlinear and perturbative
setting.
