On the volume of projections of the cross-polytope

We study properties of the volume of projections of the n-dimensional cross-polytope lozenge(n) = {x is an element of R-n vertical bar vertical bar x(1)vertical bar +...+ vertical bar x(n)vertical bar <= 1}. We prove that the projection of lozenge(n) onto a k-dimensional coordinate subspace has the maximum possible volume for k = 2 and for k = 3. We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of lozenge(n) onto a k-dimensional subspace for any n > k >= 2. (C) 2021 Elsevier B.V. All rights reserved.