The Orlicz-Petty bodies

Baocheng Zhu Hubei University for Nationalities Han Hong Memorial University of Newfoundland Deping Ye Memorial University of Newfoundland

Convex and Discrete Geometry mathscidoc:1904.40002

Int. Math. Res. Notices, 2018, 4356-4403, 2018.7
This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, under certain conditions, on the set of convex bodies in terms of the Hausdorff distance. Our proofs rely on the existence of the Orlicz-Petty bodies and the uniform boundedness of the Orlicz-Petty bodies of a convergent sequence of convex bodies. Similar results for the nonhomogeneous Orlicz geominimal surface areas are proved as well.
affine isoperimetric inequalities, affine surface area, geominimal surface area, Orlicz-Brunn-Minkowski theory, Orlicz mixed volume, Petty body
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  title={The Orlicz-Petty bodies},
  author={Baocheng Zhu, Han Hong, and Deping Ye},
  booktitle={Int. Math. Res. Notices},
Baocheng Zhu, Han Hong, and Deping Ye. The Orlicz-Petty bodies. 2018. Vol. 2018. In Int. Math. Res. Notices. pp.4356-4403.
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