Stronger versions of the Orlicz-Petty projection inequality

We verify a conjecture of Lutwak, Yang, Zhang about the equality case in the Orlicz-Petty projection inequality, and provide an essentially optimal stability version. The Petty projection inequality (Theorem 1), its Lp extension, and its analytic counterparts, the Zhang-Sobolev inequality [43] and its Lp extension by A. Cianchi, E. Lutwak, D. Yang, G. Zhang [8, 32], are fundamental affine isoperimetric and affine analytic inequalities (see in addition, e.g., D. AlonsoGutierrez, J. Bastero, J. Bernués [1], R.J. Gardner, G. Zhang [14], C. Haberl, F.E. Schuster [21, 22], C. Haberl, F.E. Schuster, J. Xio [23], E. Lutwak, D. Yang, G. Zhang [31, 33, 34], M. Ludwig [27, 28], M. Schmuckenschläger [40], F.E. Schuster, T. Wannnerer [41], J. Xiao [42]). The notion of projection body and its Lp extension have found their natural context in the work of E. Lutwak, D. Yang, G. Zhang [34], where the authors introduced the concept of Orlicz projection body. The fundamental result of [34] is the Orlicz-Petty projection inequality. The goal of this paper is to strengthen this latter inequality extending the method of E. Lutwak, D. Yang, G. Zhang [34] based on Steiner symmetrization. When the equality case of a geometric inequality is characterized, it is a natural question how close a convex body K is to the extremals if almost inequality holds for K in the inequality. Precise answers to these questions are called stability versions of the original inequalities. Stability results for geometric estimates have important applications, see for example B. Fleury, O. Guédon, G. Paouris [12] for the central limit theorem on convex bodies, ∗Supported by OTKA 75016