Converse implication from a single-radius to an all-radii small-ball estimate

Prove that, for a fixed even isotropic log-concave probability measure on R^n, a small-ball estimate at one fixed radius with an n-dimensional exponent implies small-ball estimates at all radii with the same n-dimensional exponent.

Background

The paper formulates several equivalent versions of the Slicing problem for even isotropic log-concave measures. Property (2) is an all-radii small-ball estimate of the form μ(|X| ≤ ε√n) ≤ (C_sb ε)^n for every sufficiently small ε, whereas property (3) requires such an estimate only at one fixed radius ε*√n, with μ(|X| ≤ ε√n) ≤ (C_isb ε_)^n < 1.

For a fixed measure, the paper proves the implication from the all-radii estimate to the single-radius estimate and establishes equivalences involving Gaussian-tilt formulations. It notes that a weaker all-smaller-radii estimate with exponent βn can be obtained from the single-radius bound, but that this does not recover the desired exponent n. The unresolved issue is therefore whether the single-radius estimate alone yields the full all-radii estimate for the same fixed measure.