Let \(M_{n, m}(\mathbb {R})\) be the space of \(n\times m\) real matrices. Define \(\mathscr {K}_o^{n,m}\) as the set of convex compact subsets in \(M_{n,m}(\mathbb {R})\) with nonempty interior containing the origin \(o\in M_{n, m}(\mathbb {R})\) , and \(\mathscr {K}_{(o)}^{n,m}\) as the members of \(\mathscr {K}_o^{n,m}\) containing o in their interiors. Let \(\Phi : M_{1, m}(\mathbb {R}) \rightarrow [0, \infty )\) be a convex function such that \(\Phi (o)=0\) and \(\Phi (z)+\Phi (-z)>0\) for \(z\ne o.\) In this paper, we propose the mth order Orlicz projection operator \(\Pi _{\Phi }^m: \mathscr {K}_{(o)}^{n,1}\rightarrow \mathscr {K}_{(o)}^{n,m}\) , and study its fundamental properties, including the continuity and affine invariance. We establish the related higher-order Orlicz-Petty projection inequality, which states that the volume of \(\Pi _{\Phi }^{m, *}(K)\) , the polar body of \(\Pi _{\Phi }^{m}(K)\) , is maximized at origin-symmetric ellipsoids among convex bodies with fixed volume. Furthermore, when \(\Phi \) is strictly convex, we prove that the maximum is uniquely attained at origin-symmetric ellipsoids. Our proof is based on the classical Steiner symmetrization and the Fiber symmetrization. We also investigate the special case for \(\Phi _{Q}=\phi \circ h_Q\) , where \(h_Q\) denotes the support function of \(Q\in \mathscr {K}^{1, m}_o\) and \(\phi : [0, \infty )\rightarrow [0, \infty )\) is a convex function such that \(\phi (0)=0\) and \(\phi \) is strictly increasing on \([0, \infty ).\) When \(\phi (t)=t^p\) for \(p\ge 1\) , the mth order Orlicz projection operator \(\Pi _{\Phi _Q}^m\) reduces to the special cases in the recent works [23, 24] up to a constant. We establish a higher-order Orlicz-Petty projection inequality related to \(\Pi _{\Phi _Q}^{m, *} (K)\) . Although \(\Phi _Q\) may not be strictly convex, we are able to characterize the equality under the additional assumption on \(\phi \) , such as the strict convexity of \(\phi \) .
