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Conic version of Loewner–John ellipsoid theorem

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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.

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Seeger, A., Torki, M. Conic version of Loewner–John ellipsoid theorem. Math. Program. 155, 403–433 (2016). https://doi.org/10.1007/s10107-014-0852-3

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