Ellipsoid minimizing Banach-Mazur distance to convex body

  1. Yes, it has appeared in the works of several experts in Functional Analysis. In the book Banach-Mazur distances and finite dimensional operator ideals, by N. Tomczak-Jaegermann, it is proved that in finite dimensional Banach spaces that have enough symmetries, the distance ellipsoid coincides with John's ellipsoid. In particular, it is unique. The author comments, on p.134, that in general "extremal ellipsoids may be very far from distance ellipsoids". In fact, maximal or minimal volume ellipsoids share many contact points with the (boundary of the ) corresponding convex body, whereas distance ellipsoids may share as few as only two contact points. A preprint called Remarks and examples concerning distance ellipsoids, by Dirk Praetorius, provides a result of this nature.

In this paper, the authors mention a result by B. Maurey, asserting that if a space $X$ does not have a unique (up to homothety) distance ellipsoid, then there is a subspace which has the same distance to a Hilbert space as the whole space and which has a unique distance ellipsoid. Unfortunately, Maurey never published this result. A nice corollary is that for two-dimensional Banach spaces, the distance ellipse is unique.

  1. "Banach-Mazur ellipsoid" is possible, but a terminology which is widely accepted by experts is "distance ellipsoid", e.g. - all references given above.

Tags:

Convex Geometry

Related

Homotopy pushout independent of factorization and symmetric in cofibration category Who first chose the names Alice and Bob for players A and B? Weyl's Branching Rule for $SU(N)$-Setting Is $\Bbb S^2 \times \Bbb S^4$ symplectic? An inequality related to area and sidelengths of a polygon $Area(A_1A_2....A_n) \le \frac{1}{n}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$ Is there a discrete lattice analogue of conformal transformations? Kuznetsov trace formula, orthogonality of Bessel functions If $A,B$ are upper triangular matrices such that $AX=XA\implies BX=XB$ for upper triangular $X$, is $B$ a polynomial in $A$? How to prove positivity of determinant for these matrices? primary decomposition for nonabelian cohomology of finite groups When does doubling the size of a set multiply the number of subsets by an integer? Algebraic power series over $\mathbb{F}_2$ as roots of polynomials of special form

Recent Posts