Estimations of clearance radius for finite subset of a unit ball in Rn

Let = {𝑃𝑖}, = 1,...,𝑁, be a finite set of points in R𝑛. Consider a classical problem - approximation of derivatives of function ∈ 𝐶1(R𝑛) using function values at points 𝑃𝑖. One method of solving this problem is based on building triangulation of set 𝑃. If 𝑝𝑖0, 𝑝𝑖1,..., ∈ are vertices of simplex in triangulation then we may find a function 𝑓𝑆(𝑥) = ⟨𝑎, + 𝑏, such that 𝑓(𝑝𝑖𝑘) = 𝑓𝑆(𝑝𝑖𝑘), = 0,..., 𝑛. Then we can approximate gradient ∇𝑓(𝑥) in points ∈ by gradient of linear function ∇𝑓𝑆(𝑥). By 𝑆(𝑓) denote an absolute error of gradient computation 𝑆(𝑓) = max 𝑥∈𝑆 |∇𝑓(𝑥) - ∇𝑓𝑆(𝑥)|. If simplexes diameters are getting smaller then behavior of 𝑆(𝑓) is connected not only with simplexes structure but also with triangulation geometry in whole. Let us remind that triangulation is called Delaunay triangulation if an empty sphere condition holds: for every simplex ∈ its circumsphere does not contain points of inside of itself ([4; 5]). In paper [3] and partially in paper [6] it was proven that if = 2 and a set of points is an "-net then the following estimate is true for Delaunay triangulation of max 𝑆∈𝑇 𝑆(𝑓) ≤ 𝐶(𝑓)",where constant 𝐶(𝑓) depends on the function class. V.A. Klyachin tried to get an analogous result for spaces of dimensions more than 3. But in [2] it was shown that in a multidimensional case an empty sphere condition is not enough for getting a similar estimate. Therefore in [1] a modified empty sphere condition was given. If this modified empty sphere condition holds then the following inequality is true max 𝑆∈𝑇 𝑆(𝑓) ≤ 𝐶(𝑓) · "/-𝑘,𝑛, where 𝐶(𝑓) is a constant depending on the function class, and -𝑘,𝑛 is defined in the following way. Let 𝐵(𝑥, 𝑡) be an open ball of radius > 0 with a center in ∈ R𝑛. Then -𝑘,𝑛 = inf 𝑞1,...,𝑞𝑘∈𝐵(0,1) sup 𝑦0∈𝐵(0,1){ : 𝐵(𝑦0, ) ⊂ 𝐵(0, 1) ∖ {𝑞1,..., 𝑞𝑘}}. This paper is devoted to studying -𝑘,𝑛.