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 KRnK \subset \mathbb R^n is defined as

lvr(K):=supLRnvr(K,L),\mathrm{lvr}(K):= \mathrm {sup}_{L \subset \mathbb R^n} \mathrm {vr}(K,L),

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

cnlvr(K),c \sqrt{n} \leq \mathrm{lvr}(K),

for every body KK (where c>0c > 0 is an absolute constant). This result improves the former best known lower bound, of order n/loglog(n)\sqrt{{n}/{\log \log(n)}}.

We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that lvr(K)\mathrm{lvr}(K) behaves as the square root of the dimension of the ambient space in the following cases: if KK is the unit ball of an unitary invariant norm in Rd×d\mathbb{R}^{d \times d} (e.g., the unit ball of the pp-Schatten class SpdS_p^d for any 1p1 \leq p \leq \infty), if KK is the unit ball of the full/symmetric tensor product of p\ell_p-spaces endowed with the projective or injective norm, or if KK 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