How large is the shadow of a symplectic ball?

TL;DR

This paper explores whether the volume of the orthogonal projection (shadow) of a symplectic ball onto a complex subspace always exceeds that of a standard ball, extending the classical non-squeezing theorem to middle dimensions.

Contribution

It investigates the validity of a middle-dimensional volume inequality for symplectic embeddings, generalizing Gromov's non-squeezing theorem beyond the classical case.

Findings

The volume inequality holds in the linear setting.

The inequality's validity in nonlinear and perturbative cases is analyzed.

Results suggest conditions under which the inequality is true or false.

Abstract

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.