On the volume of the Sp(n)·Sp(1) shadow of a compact set

Let F:R4n→R4n be an element of the quaternionic unitary group Sp(n) · Sp(1), let K be a compact subset of R4n, and let V be a 4. k-dimensional quaternionic subspace of R4n≅Hn. The 4. k-dimensional shadow of the image under F of K is its orthogonal projection P(F(K)) onto V. We show that there exists a 4. k-dimensional quaternionic subspace W of R4n such that the volume of the shadow P(F(K)) is the same as the volume of the section K∩. W. This is a quaternionic analogue of the symplectic linear non-squeezing result recently obtained by Abbondandolo and Matveyev.