Weak isomorphic reverse isoperimetry

Prove the weak isomorphic reverse isoperimetry conjecture: for every origin-symmetric convex body K ⊂ ℝ^n there exist a volume-preserving linear map A ∈ SL_n(ℝ) and an origin-symmetric convex body L ⊂ A K such that vol_n(L)^{1/n} ≳ vol_n(K)^{1/n} and iq(L) ≲ √n, with universal constants independent of K and n.

Background

The classical reverse isoperimetric theorem (Ball) states that every convex body K admits an affine image whose isoperimetric quotient is O(n), matching that of the regular simplex up to constants. The conjecture proposes a stronger, isomorphic variant: after an affine map and a bounded perturbation (passing to a large-volume subset), one can always achieve an isoperimetric quotient O(√n), comparable to the Euclidean ball.

The present paper proves this conjecture for the class of n-dimensional convex polytopes with O(n) facets (indeed, in a slightly translated form), but the general conjecture for all origin-symmetric convex bodies remains open.