The LYZ centroid conjecture for star bodies
For the beginning, the authors give a nice review of centroid bodies theory, in particular they formulate the LYZ Conjecture: If \( \phi \in {\mathcal C}\) and \( K \) is a star body in \( {\mathbb R }^n \) that contains the origin in its interior, then the volume ratio \( |\Gamma_{\phi} K| / |K| \) is minimized if and only if \( K \) is an ellipsoid centered at the origin. Let \( {\mathcal C} \) denote the set of convex functions \( \phi : {\mathbb R} \to [0,\infty)\) such that \( \phi (0) = 0\) and \( \phi (t) + \phi (-t) \neq 0 \) for \( t \neq 0 \), \( S^n_{o}\) be the set of star bodies in \( {\mathbb R }^n \) with respect to the origin. Let \( \Gamma_{\phi} K \) be the Orlicz centroid body for \( \phi \in {\mathcal C} \), \( x \in {\mathbb R }^n \) defined by the support function \[ h_{\Gamma_{\phi} K } (x) = \inf \left\{ \lambda > 0 : \frac{1}{|K|} \int_K \phi \left( \frac{x \cdot y}{\lambda}\right) dy \leq 1 \right\}. \] The authors study the LYZ Conjecture ``for star bodies in \( \tilde{S}^n_{o} \) without the condition of \( \phi\)'s strict convexity''. The main result of the paper is: Theorem 1.1. Let \( \phi \in {\mathcal C}\) and \( K \) be a star body with respect to the origin. Then the volume ratio \( |\Gamma_{\phi} K| / |K| \) is minimized when \( K \) is an ellipsoid centered at the origin. If \( K \in \tilde{S}^n_{o}\), then ellipsoids centered at the origin are the only minimizers. The authors reduce the statement of this theorem to the case of convex body \( \phi \) considered by E. Lutwak.
scientific article; zbMATH DE number 6916566
scientific article