General higher order \(L^p\) mean zonoids (Q6658173)

Motivated by an old generalization of the difference body, \textit{J. Haddad} et al. [``Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies'', Preprint, \url{arXiv:2304.07859}] have introduced higher-order versions of the projection body, centroid body, radial mean bodies and have generalized the corresponding inequalities. This line of research is continued here, by generalizing Zhang's mean zonoid of a convex body and its \(L^p\) version to the \(m\)th order setting. More generally, for \(n,m\in{\mathbb N}\), \(p>0\) and a fixed \(m\)-dimensional convex body \(Q\), the \((L^p,Q)\) mean zonoid of an \(n\)-dimensional convex body \(K\) is defined by \N\[\NZ_p^m(K,Q) =\left(\frac{\mathrm{vol}_{nm}(R^m_{nm+p}K)}{\mathrm{vol}_n(K)^m}\right) \Gamma_{Q,p}R^m_{nm+p}K,\N\]\Nwhere \(R^m_qK\) is the \(q\)th \(m\)th-order radial mean body of \(K\) (and hence is of dimension \(nm\)) and \(\Gamma_{Q,p}L\) is the \((L^p,Q)\) centroid body of an \(nm\)-dimensional convex body \(L\) (and hence is of dimension \(n\)). After establishing several properties of \(Z_p^m(K,Q)\), the main result says that \(\mathrm{vol}_n(Z_p^m(K,Q))/\mathrm{vol}_n(K)\) for \(p\ge 1\) attains its minimum if \(K\) is an ellipsoid. The proof is a sophisticated application of Steiner symmetrization.