On the Analog of Shephard Problem for Lp-polar Projection Bodies

MA Tong-yi

doi:10.13371/j.cnki.chin.q.j.m.2015.04.013

Chinese Quarterly Journal of Mathematics >

2015 , Vol. 30 >Issue 4: 596 - 609

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2015.04.013

On the Analog of Shephard Problem for L_p-polar Projection Bodies

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College of Mathematics and Statistics, Hexi University

MA Tong-yi(1959-), male, Huining, Gansu, a professor of Hexi University, engages in convex geometric analysis and distance geometry study.

Received date: 2014-03-21

Online published: 2020-11-19

Supported by

Supported by the National Natural Science Foundation of China(11561020,11371224)； Supported by the Science and Technology Plan of the Gansu Province(145RJZG227)；

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Abstract

For p > 0, Lutwak, Yang and Zhang introduced the concept of L_p-polar projection body Γ_-pK of a convex body K in Rⁿ. Let p ≥ 1 and K, L \subset Rⁿbe two origin-symmetric convex bodies, we consider the question of whether Γ_-p K \subset Γ_-p L implies \Omega_p（L） ≤ \Omega_p（K）,where \Omega_p（K） denotes the L_p-affine surface area of K and K = Voln（K）^-1/pK. We prove a necessary and sufficient condition of an analog of the Shephard problem for the L_p-polar projection bodies.

Key words： convex body; L_p-polar projection body; L_p-affine surface area; Fourier transform; Shephard problem

Cite this article

MA Tong-yi . On the Analog of Shephard Problem for L_p-polar Projection Bodies[J]. Chinese Quarterly Journal of Mathematics, 2015 , 30(4) : 596 -609 . DOI: 10.13371/j.cnki.chin.q.j.m.2015.04.013

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