Entropy and asymptotic geometry of non-symmetric convex bodies. (English) Zbl 0974.52004
A short version of this paper (without proofs) appeared in [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 4, 303-308 (1999; Zbl 0964.52006)].
We quote from the corresponding review: The authors extend a number of basic results in the local theory of Banach spaces from the setting of centrally symmetric convex bodies to the non-symmetric case. A main tool are inequalities describing the volume behaviour around the centroid of a convex body. Applications given concern covering numbers (entropy) of convex bodies and their polars, a non-symmetric version of the quotient subspace theorem, and existence and properties of \(M\)-ellipsoids of non-symmetric convex bodies.
We quote from the corresponding review: The authors extend a number of basic results in the local theory of Banach spaces from the setting of centrally symmetric convex bodies to the non-symmetric case. A main tool are inequalities describing the volume behaviour around the centroid of a convex body. Applications given concern covering numbers (entropy) of convex bodies and their polars, a non-symmetric version of the quotient subspace theorem, and existence and properties of \(M\)-ellipsoids of non-symmetric convex bodies.
Reviewer: Wolfgang Weil (Karlsruhe)
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 46B07 | Local theory of Banach spaces |
Citations:
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