General mixed chord-integrals of star bodies

Abstract

Mixed chord-integrals of star bodies were first defined by Lu. In this paper, the concept of mixed chord-integrals is extended to general mixed chord-integrals, which is motivated by the recent work on general $L_p$-affine isoperimetric inequalities by Haberl, et al. For this new notion of general mixed chord-integrals, isoperimetric and Aleksandrov-Fenchel type inequalities are established which generalize inequalities obtained by Lu, and a cyclic inequality is also obtained. Furthermore, we prove several Brunn-Minkowski type inequalities for general mixed chord-integrals.