关于Lp-相交体的Busemann-Petty型问题

数学季刊 ›› 2012, Vol. 27 ›› Issue (2): 259-269.

关于L_p-相交体的Busemann-Petty型问题

School of Mathematics and Statistics, Hexi University

收稿日期:2010-10-08 出版日期:2012-06-30 发布日期:2023-03-29
作者简介:MA Tong-yi(1959-), male, native of Huining, Gansu, a professor of Hexi University, engaged in convex geometric analysis and distance geometry study.
基金资助:
Supported by the NNSF of China(11161019)； Supported by the Foundation of the Education Department of Gansu Province(1009B-09)

On the Busemann-Petty’s Problem for L_p-intersection Body

School of Mathematics and Statistics, Hexi University

Received:2010-10-08 Online:2012-06-30 Published:2023-03-29
About author:MA Tong-yi(1959-), male, native of Huining, Gansu, a professor of Hexi University, engaged in convex geometric analysis and distance geometry study.
Supported by:
Supported by the NNSF of China(11161019)； Supported by the Foundation of the Education Department of Gansu Province(1009B-09)

摘要/Abstract

摘要： Haberl and Ludwig introduced the L_p-intersection body I_pK for an origin-symmetric star body K in R_n, where p < 1 and p ≠ 0. In this paper, we consider the Busemann-Petty’s problem for L_p-intersection bodies I_pK and I_pL. That is, whether I_pK I_pL implies Vol_n(K) ≤ Vol_n(L). We obtain that for two origin-symmetric star bodies K and L in R_n, such that (R_n, ||·||K) embeds in L_p and I_pK I_pL, then vol_n(K) ≤ vol_n(L) for 0 < p < 1 and vol_n(K) ≥ vol_n(L) for p < 0.

关键词: star body, L_p-intersection bodies, isometrically embedded in L_p, generalized cosine transforms

Abstract: Haberl and Ludwig introduced the L_p-intersection body I_pK for an origin-symmetric star body K in R_n, where p < 1 and p ≠ 0. In this paper, we consider the Busemann-Petty’s problem for L_p-intersection bodies I_pK and I_pL. That is, whether I_pK I_pL implies Vol_n(K) ≤ Vol_n(L). We obtain that for two origin-symmetric star bodies K and L in Rn, such that (Rn, ||·||K) embeds in L_p and I_pK IpL, then vol_n(K) ≤ vol_n(L) for 0 < p < 1 and vol_n(K) ≥ vol_n(L) for p < 0.

Key words: star body, L_p-intersection bodies, isometrically embedded in L_p, generalized cosine transforms

中图分类号:

O186.5

马统一. 关于L_p-相交体的Busemann-Petty型问题[J]. 数学季刊, 2012, 27(2): 259-269.

MA Tong-yi. On the Busemann-Petty’s Problem for L_p-intersection Body[J]. Chinese Quarterly Journal of Mathematics, 2012, 27(2): 259-269.

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