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Mathematical Programming

, Volume 155, Issue 1–2, pp 403–433 | Cite as

Conic version of Loewner–John ellipsoid theorem

  • Alberto SeegerEmail author
  • Mounir Torki
Full Length Paper Series A

Abstract

We extend John’s inscribed ellipsoid theorem, as well as Loewner’s circumscribed ellipsoid theorem, from convex bodies to proper cones. To be more precise, we prove that a proper cone \(K\) in \(\mathbb {R}^n\) contains a unique ellipsoidal cone \(Q^\mathrm{in}(K)\) of maximal canonical volume and, on the other hand, it is enclosed by a unique ellipsoidal cone \(Q^\mathrm{circ}(K)\) of minimal canonical volume. In addition, we explain how to construct the inscribed ellipsoidal cone \(Q^\mathrm{in}(K)\). The circumscribed ellipsoidal cone \(Q^\mathrm{circ}(K)\) is then obtained by duality arguments. The canonical volume of an ellipsoidal cone is defined as the usual \(n\)-dimensional volume of a certain truncation of the cone.

Keywords

Convex body Convex cone Inscribed ellipsoidal cone Circumscribed ellipsoidal cone 

Mathematics Subject Classification

51M25 52A38 47L07 

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité d’AvignonAvignonFrance
  2. 2.CERIUniversité d’AvignonAvignonFrance

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