Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (1): 180-188.doi: 10.1007/s10473-025-0114-x

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ON LEGENDRE AND LYZ ELLIPSOIDS

Xinbao Lu1, Ge Xiong1,*, Jiangyan Tao2   

  1. 1. School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai 200092, China;
    2. School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China
  • Received:2024-09-04 Published:2025-02-06
  • Contact: *Ge Xiong, E-mail,: xiongge@tongji.edu.cn
  • About author:Xinbao Lu, E-mail,: xinbaolu@tongji.edu.cn; Jiangyan Tao, E-mail,: taojiangyan@zjnu.edu.cn
  • Supported by:
    Xiong's work was supported by the NSFC (12271407).

Abstract: We study the Legendre ellipsoid and the LYZ ellipsoid. First, we give a direct proof of the Cramer-Rao inequality for convex bodies. Second, we prove that origin-centered ellipsoids are the only convex bodies with identical Löwner and Legendre ellipsoids. Finally, we establish a mean width inequality for convex bodies whose LYZ ellipsoids are the Euclidean unit ball.

Key words: Legendre ellipsoid, LYZ ellipsoid, Löwner ellipsoid

CLC Number: 

  • 52A40
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