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.
Xinbao Lu
,
Ge Xiong
,
Jiangyan Tao
. ON LEGENDRE AND LYZ ELLIPSOIDS[J]. Acta mathematica scientia, Series B, 2025
, 45(1)
: 180
-188
.
DOI: 10.1007/s10473-025-0114-x
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