Dyn–Farkhi subadditivity when A=B in higher dimensions

Determine whether, for every compact set A ⊂ R^n with n ≥ 3, the inequality d^2(A+A) ≤ 2 d^2(A) holds, where d(A) denotes the Hausdorff distance from A to conv(A).

Background

The Dyn–Farkhi conjecture posits that d^2 is subadditive on compact sets: d^2(A+B) ≤ d^2(A) + d^2(B). This paper proves the conjecture in two dimensions and recalls that Cassels' effective standard deviation v^2 is subadditive and that d(A) ≤ rad(A), motivating the conjecture.

Fradelizi et al. showed the conjecture is false in dimension n ≥ 3 for general pairs of sets, while it is true in dimension n = 1. Despite this, the special case A = B has not been resolved in higher dimensions; this paper emphasizes that this remains open even though the general conjecture is false.