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

Let F be an element of the quaternionic unitary group Sp(n)Sp(1), let K be a compact subset of Euclidean 4n-space, and let V be a 4k-dimensional quaternionic subspace of Euclidean 4n-space identified with the (right) quaternionic vector space of column n-tuples of quaternions.The 4k-dimensional shadow of the image under F of K is its orthogonal projection P(F(K)) onto V. We show that there exists a 4k-dimensional quaternionic subspace W of Euclidean 4n-space such that the volume of the shadow P(F(K)) is the same as the volume of the section of K by W. This is a quaternionic analogue of the symplectic linear non-squeezing result recently obtained by Abbondandolo and Matveyev