Conic version of Loewner-John ellipsoid theorem

• 成果类型: Article
• 署名作者: Seeger, Alberto; Torki, Mounir
• 署名单位: Avignon Universite; Avignon Universite
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0852-3
• 发表日期: 2016
• 页码: 403-433
• 关键词: closed convex cones; pointedness

摘要

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 in contains a unique ellipsoidal cone of maximal canonical volume and, on the other hand, it is enclosed by a unique ellipsoidal cone of minimal canonical volume. In addition, we explain how to construct the inscribed ellipsoidal cone . The circumscribed ellipsoidal cone is then obtained by duality arguments. The canonical volume of an ellipsoidal cone is defined as the usual -dimensional volume of a certain truncation of the cone.