Optimality of the exponents in the quantitative uniqueness bound

Determine whether the exponent 1−1/(1+α) in Theorem 4 (which bounds the L∞-diameter of the set of Kantorovich potentials for costs c ∈ C^{1,α} in terms of the Hausdorff distance between supp ρ and a connected set) is optimal. Equivalently, for p-costs c(x,y)=||x−y||^p on compact sets, determine whether the implied exponents 1/2 (for p≥2) and 1−1/p (for 1<p≤2) are optimal.

Background

Theorem 4 proves an L∞-diameter bound of the form C(1+C_Ω)^2(d_H^{1−1/(α+1)}+d_H) when c ∈ C^{1,α} and Ω satisfies a uniform bounded-curvature connectivity condition. Specializing to p-costs on compact domains yields exponents 1/2 (p≥2) and 1−1/p (1<p≤2).

The authors note they can rule out improvements beyond exponent 1 in general via a grid example, but the optimality of the specific exponents arising from C^{1,α} regularity remains unresolved.