Orlicz mixed chord-integrals

Chang-Jian Zhao; Chang-Jian Zhao

doi:10.3934/math.2020427

AIMS Mathematics

2020, Volume 5, Issue 6: 6639-6656. doi: 10.3934/math.2020427

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Research article

Orlicz mixed chord-integrals

Chang-Jian Zhao ^,

Department of Mathematics, China Jiliang University, Hangzhou 310018, P. R. China

Received: 02 July 2020 Accepted: 03 August 2020 Published: 25 August 2020
MSC : 46E30, 52A30, 52A40

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.
- star body,
- first order variation,
- chord integral,
- mixed chord integral,
- $L_{p}$-mixed chord integral,
- Orlicz chord addition,
- Orlicz mixed chord integral
Citation: Chang-Jian Zhao. Orlicz mixed chord-integrals[J]. AIMS Mathematics, 2020, 5(6): 6639-6656. doi: 10.3934/math.2020427

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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.

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