The Multiple Radial Blaschke–Minkowski Homomorphisms

In the paper, our main aim is to generalize the mixed radial Blaschke–Minkowski homomorphisms and Aleksandrov–Fenchel inequality for mixed radial Blaschke–Minkowski homomorphisms to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating Orlicz first order variation of dual quermassintegrals of the mixed radial Blaschke–Minkowski homomorphisms and call it Orlicz multiple radial Blaschke–Minkowski homomorphisms. The fundamental notions and conclusions of dual quermassintegrals of mixed radial Blaschke–Minkowski homomorphisms and Aleksandrov–Fenchel inequality for mixed radial Blaschke–Minkowski homomorphisms are extended to an Orlicz setting. The related concepts and inequalities of Orlicz mixed intersection bodies are also derived. The new Orlicz–Aleksandrov–Fenchel inequality for dual quermassintegrals of Orlicz multiple radial Blaschke–Minkowski homomorphisms in special case yield not only new $L_p$ type Aleksandrov–Fenchel inequality and Orlicz–Minkowski inequality but also Orlicz–Aleksandrov–Fenchel inequality for Orlicz mixed intersection bodies.