Abstract
In this paper, the extremal problem, min{( ΦK) :o ∈ ΦK Φ L, Φ ∈ GL(n)} , of two convex bodies K and L in is considered. For K to be in extremal position in terms of a decomposition of the identity, give necessary conditions together with the optimization theorem of John. Besides, we also consider the weaker optimization problem: min{((ΦK))p : ΦK Φ B2n, ΦK ∩ Sn-1 ≠ Φ, Φ ∈ GL(n)} . As an application, we give the geometric distance between the unit ball B2n and a centrally symmetric convex body K.
Highlights
IntroductionNotice that the study of the classical John theorem went back to John [2]
Let γ n be the classical Gaussian probability measure with density ( )n − | x|2 e2, and|| ⋅ ||K is the Minkowski functional of a convex body K ⊂ nAn important quantity on local theory of Banach space is the associated l-norm: l(K ) = ∫ n || x ||K dγ n (x).The minimum of the functional
For K to be in extremal position in terms of a decomposition of the identity, give necessary conditions together with the optimization theorem of John
Summary
Notice that the study of the classical John theorem went back to John [2] It states that each convex body K contains a unique ellipsoid of maximal volume, and when B2n is the maximal ellipsoid in K, it can be characterized by points of contact between the boundary of K and that of B2n. It was provided in [5] that a generalization of John’s theorem for the maximal volume position of two arbitrary smooth convex bodies. For p ≥ 1 , denote lp -norm by ( ) ∫ lp (K ) = n || x ||Kp dγ n (x) p
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
