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On the symmetric average of a convex body
-
Olivier Guédon
and
A. E. Litvak
Published/Copyright:
November 11, 2011
Abstract
We introduce a new parameter, symmetric average, which measures the asymmetry of a given non-degenerated convex body K in 
. Namely, sav(K) = infa∈int K ∫Ka ‖– x‖Kadx/|K|, where |K| denotes the volume of K and Ka = K – a. We show that for polytopes sav(K) ≤ C ln N, where N is the number of facets of K. Moreover, in general 
and equality in the lower bound holds if and only if K is centrally symmetric. We apply these estimates to provide bounds for covering K by homothets of K ∩ –K.
Received: 2009-07-09
Published Online: 2011-11-11
Published in Print: 2011-November
© de Gruyter 2011
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Articles in the same Issue
- Quasigeodesics and farthest points on convex surfaces
- The geometry of canal surfaces and the length of curves in de Sitter space
- On the mutual position of two irreducible conics in PG(2, q), q odd
- On the symmetric average of a convex body
- Ample vector bundles and polarized manifolds of sectional genus three
- Flat Laguerre planes of Kleinewillinghöfer type III.B
- Quotients of hypersurfaces in weighted projective space
- Characterizing the mixed volume
- Uniqueness of lattice packings and coverings of extreme density
- On the classification of convex lattice polytopes
- Blichfeldt-type inequalities and central symmetry
- Valuations on function spaces
