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Inscribed and circumscribed ellipsoidal cones: volume ratio analysis

  • Alberto SeegerEmail author
  • Mounir Torki
Original Paper
  • 55 Downloads

Abstract

Let K be a pointed closed convex cone with nonempty interior in \(\mathbb {R}^n\). The inscribed ellipsoidal cone in K, denoted by \(E^{\uparrow }(K)\), is defined as the unique ellipsoidal cone that maximizes the volume among all ellipsoidal cones contained in K. The cone \(E^{\uparrow }(K)\) can be viewed as an optimal inner approximation of K. One of the goals of this work is to analyze which percentage of the volume of K is captured by \(E^{\uparrow }(K)\). In particular, we show that the lowest possible percentage occurs when K is a simplicial cone. Our result corresponds to a conic version of a celebrated theorem of Ball (J Lond Math Soc 44:351–359, 1991) on the volume ratio function of a convex body. The problem of analyzing the quality of the outer approximation of K by means of its circumscribed ellipsoidal cone can be treated by using duality arguments.

Keywords

Convex cone Inscribed ellipsoidal cone Circumscribed ellipsoidal cone Volume ratio Axial symmetry 

Mathematics Subject Classification

51M25 52A38 47L07 

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Copyright information

© The Managing Editors 2018

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité d’AvignonAvignonFrance
  2. 2.Université d’Avignon, CERIAvignonFrance

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