On Properties of a Regular Simplex Inscribed into a Ball

Abstract

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$.

Linear Interpolation on a Euclidean Ball in Rⁿ

“…were found in [4]. Namely, assume k n coincides with that of the numbers a n and a n + 1 which delivers to ψ(t) the maximum value.…”

“…As a conjecture, Theorem 1 was formulated by the author in [4]. Also in [4], the statement of Theorem 1 in case m = 1 was proved in another way.…”

“…For 1 ≤ n ≤ 4, equality (9) and characterization of minimal projectors were obtained in [3] and [4] by another methods.…”

Geometric Estimates in Interpolation on an n-Dimensional Ball

“…As a conjecture, Theorem 8.1 was formulated by the author in [29] . Also in [29], the statement of the theorem for m = 1 was proved in a simpler way.…”

“…As a conjecture, Theorem 8.1 was formulated by the author in [29] . Also in [29], the statement of the theorem for m = 1 was proved in a simpler way. In the case m = 1 the points y have the form y ( j) = 2c − x ( j) , j = 1, .…”