Geometric Estimates in Interpolation on an n-Dimensional Ball

Suppose n ∈ N . Let Bₙ be the Euclidean unit ball in Rⁿ given by the inequality \| x \| 1 , \| x \|: = ( ∑i = 1ⁿ xᵢ² )1/2 . By C(Bₙ) we mean a space of continuous functions f:Bₙ → R with the norm \| f \|_C(Bₙ): = max_x ∈ Bₙ | f(x) | . The symbol Π ₁( Rⁿ ) denotes a set of polynomials in n variables of degree 1 , i. e., a set of linear functions upon Rⁿ . Assume x^(1), … ,x^(n + 1) are the vertices of an n -dimensional nondegenerate simplex S ⊂ Bₙ . The interpolation projector P:C(Bₙ) → Π ₁(Rⁿ) corresponding to S is defined by the equalities Pf( x^(j) ) operator from C(Bn) onto C(Bₙ) . Let us define θ ₙ(Bₙ) as the minimal value of \| P \|_Bₙ under the condition x^(j) ∈ Bₙ . We describe the approach in which the projector’s norm can be estimated from below via the volume of the simplex. Let χ ₙ(t): = 12ⁿn! [ (t² - 1)ⁿ ]⁽ⁿ⁾ be the standardized Legendre polynomial of degree n . We prove that \| P \|_Bₙ χ ₙ^ - 1( vol(Bₙ)vol(S) ). From this, we obtain the equivalence θ ₙ(Bₙ) √ n . Also we estimate the constants from such inequalities and give the comparison with the similar relations for linear interpolation upon the n -dimensional unit cube. These results have applications in polynomial interpolation and computational geometry.