Orlicz mixed chord-integrals

Abstract

In this paper, we introduce an affine geometric quantity and call it Orlicz mixed chord integral by defining a new Orlicz chord addition, which generalizes the mixed chord integrals to Orlicz space. The Minkoswki and Brunn-Minkowski inequalities for the Orlicz mixed chord integrals are established. The new inequalities in special cases yield $L_{p}$-Minkowski and Brunn-Minkowski inequalities for the chord integrals. The related concepts and inequalities of $L_{p}$-mixed chord integrals are derived. As an application, a new isoperimetric inequality for the chord integrals is given. As extensions, Orlicz multiple mixed chord integrals and Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals are also derived here for the first time.

Keywords

• star body
• first order variation
• chord integral
• mixed chord integral
• $l_{p}$-mixed chord integral
• orlicz chord addition
• orlicz mixed chord integral