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Sections and Projections of \(L_p\)-Zonoids and Their Polars

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Abstract

Let \(1\le k\le n\). Sharp upper and lower bounds for the volume of k-dimensional projections (or sections) of \(L_p\)-zonoids (or their polars) with even isotropic generating measures are established. As special cases, sharp volume inequalities for k-dimensional sections and projections of \(\ell _p^n\)-balls are recovered. The necessary conditions of equalities are new.

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Acknowledgements

The authors are indebted to the referees for many valuable suggestions and comments. The first author was supported by NSFC-Henan Joint Fund (No. U1204102) and Key Research Project for Higher Education in Henan Province (No. 17A110022). The second author was supported in part by the National Natural Science Foundation of China (No. 11626115 and 11371239). The third author was supported by the National Natural Science Foundation of China (No. 11601310) and Shanghai Sailing Program (No. 16YF1403800).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China

    Ai-Jun Li

  2. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

    Qingzhong Huang

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Dongmeng Xi

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  1. Ai-Jun Li
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  2. Qingzhong Huang
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  3. Dongmeng Xi
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Correspondence to Ai-Jun Li.

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Li, AJ., Huang, Q. & Xi, D. Sections and Projections of \(L_p\)-Zonoids and Their Polars. J Geom Anal 28, 427–447 (2018). https://doi.org/10.1007/s12220-017-9827-y

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  • DOI: https://doi.org/10.1007/s12220-017-9827-y

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