A new affine invariant for polytopes and Schneider's projection problem (Q2701680)

If \(P\) is a convex polytope in \(\mathbb{R}^n\) which contains the origin in its interior, and \(u_1,\dots,u_N\) are the outer normal unit vectors to the faces of \(P\), with \(h_1,\dots,h_N\) the corresponding distances of the faces from the origin and \(a_1,\dots,a_N\) the corresponding areas of the faces, then let NEWLINE\[NEWLINEU(P)^n= {1\over n^n}\sum_{u_{i_1} \wedge\cdots \wedge u_{i_n}\neq 0}h_{i_1} \cdots h_{i_n} a_{i_1} \cdots a_{i_n}.NEWLINE\]NEWLINE The new functional \(U\) thus defined for convex polytopes is invariant under volume preserving linear maps.NEWLINENEWLINENEWLINEThe authors prove that suitable combinations of this functional and the volume provide sharp upper bounds for the volume of the projection body of a convex polytope \(P\), provided that \(P\) has the origin as centre of symmetry or as its John point. The equality cases characterize parallelotopes and simplices, respectively. Since the new functional is always smaller than the volume (but close to it for polytopes with many facets), one can deduce (non-sharp) upper bounds for the volume of the projection body in terms of the volume. This is a contribution to the open problem of finding exact bounds in this situation.NEWLINENEWLINENEWLINEThe paper contains a series of further inequalities, including certain generalizations of Keith Ball's reverse isoperimetric inequality.