The \(L_p\) John ellipsoids for general measures. (English) Zbl 1505.52004
The classical John ellipsoid of a convex body \(K\subset{\mathbb R}^n\) is the (unique) ellipsoid of maximal volume that is contained in \(K\). E. Lutwak et al. [Proc. Lond. Math. Soc. (3) 90, No. 2, 497–520 (2005; Zbl 1074.52005)] have introduced the \(L_p\) John ellipsoids for \(0<p\le\infty\), considering only convex bodies containing the origin in the interior, admitting only origin-symmetric ellipsoids, and replacing the condition \(E\subseteq K\) by the condition that some integral, defined via the \(L_p\) Brunn-Minkowski theory and depending on \(E\) and \(K\), is at most \(1\). This is here further generalized. Let \(\mu\) be a Borel measure on \({\mathbb R}^n\) having a density \(g\) (with respect to Lebesgue measure) that is continuous on the support of \(\mu\). For \(p>0\) and a convex body \(K\) containing the origin in the interior, Problem \(M_{\mu,p}\) then asks to maximize the volume of the origin-symmetric ellipsoids \(E\) under the side condition that a certain integral, coming from the \(L_p\) Brunn-Minkowski theory and involving \(E\), \(K\) and \(g\), is at most \(1\). This problem is solved in the present paper, and it is shown that the maximizing ellipsoid has properties generalizing those of the \(L_p\) John ellipsoids. Special attention is paid to the Gaussian measure.
Reviewer: Rolf Schneider (Freiburg im Breisgau)
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
Citations:
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