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Davis’ convexity theorem and extremal ellipsoids. (English) Zbl 1202.52003
The authors obtain several uniqueness results for the maximal/minimal ellipsoids contained in/containing a convex body, with respect to size functions different from the usual volume.
More precisely, a function $$f:{\mathbb R}^d_{\geq}\longrightarrow{\mathbb R}_{\geq}$$ is called a size function for an ellipsoid if it is continuous, monotone strictly increasing in all its arguments and symmetric. Then, denoting by $$\omega^p:{\mathbb R}^d\longrightarrow{\mathbb R}^d$$ the function given by $$\omega^p(x_1,\dots,x_d)^{\intercal}=\bigl(|x_1|^p,\dots,|x_d|^p)^{\intercal}$$, the authors show that if $$F\subset{\mathbb R}^d$$ is a convex body and $$f$$ is a size function for ellipsoids such that $$f\circ\omega^1$$ (resp. $$f\circ\omega^{-1}$$) is strictly concave on $${\mathbb R}^d_{\geq}$$ (strictly convex on $${\mathbb R}^d_{>}$$), then among all ellipsoids contained in (containing) $$F$$ there exists a unique ellipsoid which is maximal (minimal) with respect to $$f$$. Analogous results are obtained when $$p=1/2$$ and $$p=-1/2$$.
C. Davis’ theorem [Arch. Math. 8, 276–278 (1957; Zbl 0086.01702)] on convex invariant functions of hermitian matrices is one of the main tools in the proofs of these theorems. The authors also provide examples of convex bodies and size functions with non-unique maximal or minimal ellipsoids.

MSC:
 52A27 Approximation by convex sets 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)
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