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(p, q)-John Ellipsoids

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Abstract

As an extension of the classical John ellipsoid and the \(L_{p}\)-John ellipsoids due to Lutwak–Yang–Zhang, this paper studies (p, q)-John ellipsoids. We consider an optimization problem about the (p, q)-mixed volumes, whose solution is uniquely existed for all \(0<p\le q\). The solution allows us to introduce the concept of (p, q)-John ellipsoids. As applications, we established an analog of the John’s inclusion theorem and Ball’s volume-ratio inequality for (p, q)-John ellipsoids. Moreover, the connection between the isotropy of measures and the characterization of (p, q)-John ellipsoids is demonstrated.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Hexi University, Zhangye, 734000, Gansu, People’s Republic of China

    Tongyi Ma & Yibin Feng

  2. College of Science, Northwest Agriculture and Forestry University, Yangling, 712100, Shaanxi, People’s Republic of China

    Denghui Wu

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  1. Tongyi Ma
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  2. Denghui Wu
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Correspondence to Denghui Wu.

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This work was supported by the National Natural Science Foundation of China (Grant No. 11561020) and Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2020JQ-236)

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Ma, T., Wu, D. & Feng, Y. (p, q)-John Ellipsoids. J Geom Anal 31, 9597–9632 (2021). https://doi.org/10.1007/s12220-021-00621-4

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