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Conic version of Loewner–John ellipsoid theorem.

Abstract : We extend John's inscribed ellipsoid theorem, as well as Loewner's circum-scribed ellipsoid theorem, from convex bodies to proper cones. To be more precise, we prove that a proper cone K in R n contains a unique ellipsoidal cone Q in (K) of maximal canonical volume and, on the other hand, it is enclosed by a unique ellipsoidal cone Q circ (K) of minimal canonical volume. In addition, we explain how to construct the inscribed ellipsoidal cone Q in (K). The circumscribed ellipsoidal cone Q 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.
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Submitted on : Friday, June 3, 2016 - 11:13:11 AM
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Alberto Seeger, Mounir Torki. Conic version of Loewner–John ellipsoid theorem.. Mathematical Programming, Series A, Springer, 2014, ⟨10.1007/s10107-014-0852-3⟩. ⟨hal-01326160⟩



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