Optimal constant-set and exponentially small-set scales under unconditional regularity

Determine whether, under unconditional Q-regularity with inner and outer copies x_0+rQ\subset K\subset y_0+RQ, the constant-set \ell_0-isoperimetric scale is of order r/(nR) and the exponentially small-set scale is of order r/R.

Background

The paper proves a lower bound for the \ell_0 boundary of subsets of a convex body that is regular with respect to an unconditional convex body Q. For constant-sized subsets, the lower bound is of order r/(n^2R), while the constructed upper bound is of order r/(nR). For exponentially small subsets, the construction has boundary ratio of order r/R, whereas the lower bound has an additional factor depending on n and the small-set profile.

The authors conjecture that the upper-bound constructions capture the correct behavior at the extreme volume scales. Establishing this would improve the main lower bound by a factor of n and would make the result sharp, at least for constant-sized and exponentially small subsets.