Universal monotonicity of support scaling exponents in dimensions d ≥ 3

Determine whether the upper support scaling exponents of a compact set A ⊂ R^d are universally nondecreasing in the index for all d ≥ 3; that is, prove or refute that s_0(A) ≤ s_1(A) ≤ ··· ≤ s_{d−1}(A) always holds when d ≥ 3.

Background

In R^2, the support scaling exponents satisfy s_0(A) ≤ s_1(A) with s_1(A) = D_M(A). The authors discuss how, under equality conditions in \eqref{eq:s<max}, one would obtain a nondecreasing chain s_0(A) ≤ ··* ≤ s_{d−1}(A).

However, in higher dimensions the general validity of this monotonic chain is unknown, and the authors explicitly label it an unresolved open problem.