Abstract
<abstract> In the paper, our main aim is to generalize the mixed affine quermassintegrals of <italic>j</italic> convex bodies to the Orlicz space. We find a new affine geometric quantity by calculating first-order variation and call it <italic>Orlicz multiple affine quermassintegrals</italic>. The mixed affine quermassintegrals and AleksandrovFenchel inequality for the mixed affine quermassintegrals of <italic>j</italic> convex bodies are extended to an Orlicz setting. A new Orlicz-Aleksandrov-Fenchel inequality for the mixed affine quermassintegrals of <italic>j</italic> convex bodies is established. The new Orlicz-Aleksandrov-Fenchel inequality in special cases yield the classical Aleksandrov-Fenchel inequality for mixed volumes, the Aleksandrov-Fenchel inequality for the mixed affine quermassintegrals which is just built, and Zou's Orlicz Minkowski inequality for affine quermassintegrals, respectively. This new concept of <italic>L<sub>p</sub></italic>-multiple affine quermassintegrals and <italic>L<sub>p</sub></italic>-AleksandrovFenchel inequality for the <italic>L<sub>p</sub></italic>-multiple affine quermassintegrals is also derived. Moreover, the Orlicz multiple mixed volumes and the Orlicz-AleksandrovFenchel inequality for the mixed volumes are also included in our new conclusions. As an application, a new Orlicz-Brunn-Minkowski inequality for the mixed affine quermassintegrals of <italic>j</italic> convex bodies is proved. </abstract>
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