New results on affine invariant points

We prove a conjecture of B. Grünbaum stating that the set of affine invariant points of a convex body equals the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof of the fact that the affine space of affine linear points is infinite...

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Bibliographic Details
Published in:Israel journal of mathematics 2017-04, Vol.219 (2), p.529-548
Main Author: Mordhorst, Olaf
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Ellipsoids
Group Theory and Generalizations
Invariants
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
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Summary:We prove a conjecture of B. Grünbaum stating that the set of affine invariant points of a convex body equals the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof of the fact that the affine space of affine linear points is infinite dimensional. In particular, we show that the set of affine invariant points with no dual is of the second category. We investigate extremal cases for a class of symmetry measures. We show that the centers of the John and Löwner ellipsoids can be far apart and we give the optimal order for the extremal distance between the two centers.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-017-1489-8