Ellipsoid-growth trace conjecture for symmetric convex bodies

Determine whether, for every symmetric convex body K in R^n and every origin-centered ellipsoid E such that the covariance matrix of the uniform measure on K intersected with E satisfies A_{K∩E}=I_n, the covariance trace of the uniform measure on K intersected with 2E is bounded by a quantity of order n.

Background

The paper discusses a proposed strategy for rounding convex bodies that repeatedly enlarges an ellipsoid while preserving near-isotropy of its intersection with the body. The cited conjecture asserts that, when the current intersection is exactly isotropic, doubling the ellipsoid keeps the enlarged intersection sufficiently well-rounded, at least in the weaker sense that its covariance trace is of order n.

The paper constructs a symmetric counterexample showing that this conjecture is false: after an appropriate constant enlargement, the covariance trace can be of order n squared, and the covariance can fail the upper near-isotropy bound in the original coordinates. Thus, the conjecture is explicitly stated in the paper but is resolved negatively by the paper’s own counterexample.