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<p>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.</p>

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On Legendre and LYZ ellipsoids

  • 期刊论文

  • 发表日期：

    2025年1月24日

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  • Xinbao Lu,
  • Ge Xiong,
  • Jiangyan Tao

摘要

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.