It is known that given a set of Areas $A\_f$ and normals $\vec{n}\_f$ if $\sum\_f A\_f \vec{n}\_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov book on Convex Polyhedra).


Obviously here I'm identifying all the isometric polyhedra.


In principle with the same set of areas and normals one can build "others" polyhedra if we relax the convexity requirement. 


What I want to prove is that in the collection of all the possible polyhedra one can build from a given set of Areas $A\_f$ and normals $\vec{n}\_f$ the convex one is the one with bigger volume.


Thank you for your help.
Pietro



Volumes of convex vs non-convex polyhedra with prescribed facets areas
The convex polytope has the largest volume. This was proved in


K. Boroczky, I. Bárány, E. Makai Jr. & J. Pach: Maximal volume enclosed by plates and proof of the chessboard conjecture, Discrete Math. 69 (1986) 101–120.


For measures and convex bodies, a proof is provided by


Zhang, Gaoyong: The affine Sobolev inequality. J. Differential Geom. 53 (1999), no. 1, 183–202. 


