Reverse Triangle Inequalities for Riesz Potentials and Connections ...

For the unit ball we have the following result.Proposition 3.4. For the unit ball B N in R N there holds, for all −∞ < α < N,1 − M N−αm (B N )mMoreover the following asymptotic formulas hold as m → ∞ :≥ C δ B(α, m) ≥ 2 α−N − M m N−α (B N ). (15)Nm⎧( ) (N−α)/N⎪⎨Γ(1 + N/2)C δ −σ(N − α, N)m −α/N , 0 > α > −∞ ,B(α, m) ∼π N⎪⎩N/2 − log m , α = 0 ,(16)where σ(N − α, N) is a positive constant that depends only on α and N.We remark that asymptotic formulas similar to those in Proposition 3.4 can be obtainedfor C δ E (α, m) for a large class of N-dimensional subsets of RN by appealing to the results in[4] and [5].4 ProofsWe begin with a lemma that will be used in the proof of Theorem 2.3. Let F n = {x k,n } n k=1be a set of n points in E. Let τ n be their normalized counting measure and let 0 < α < N.We define the discrete α-energy of τ n byE α [τ n ] :=2n(n − 1)∑1≤j