Asymptotic estimates for the largest volume ratio of a convex body

  • Daniel Galicer

    Universidad de Buenos Aires, Argentina
  • Mariano Merzbacher

    Universidad de Buenos Aires, Argentina
  • Damián Pinasco

    Universidad T. Di Tella, Buenos Aires, Argentina
Asymptotic estimates for the largest volume ratio of a convex body cover
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Abstract

The largest volume ratio of a given convex body is defined as

where the sup runs over all the convex bodies . We prove the following sharp lower bound:

for every body (where is an absolute constant). This result improves the former best known lower bound, of order .

We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that behaves as the square root of the dimension of the ambient space in the following cases: if is the unit ball of an unitary invariant norm in (e.g., the unit ball of the -Schatten class for any ), if is the unit ball of the full/symmetric tensor product of -spaces endowed with the projective or injective norm, or if is unconditional.

Cite this article

Daniel Galicer, Mariano Merzbacher, Damián Pinasco, Asymptotic estimates for the largest volume ratio of a convex body. Rev. Mat. Iberoam. 37 (2021), no. 6, pp. 2347–2372

DOI 10.4171/RMI/1263