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~continuous functions $f:B\to \mathbb{R}$ with the uniform norm $\Vert f{\Vert }_{C(B)}:={\max }_{x\in B}|f(x)|.$ By ${\Pi }_1\left({\mathbb{R}}^n\right)$ we mean a set of polynomials of degree $\le 1$, 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 $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,$ $\dots ,$ $n+1$. The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\Vert P{\Vert }_B={\max }_{x\in B}\sum |{\lambda }_j(x)|.$ 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 \linebreak of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\Vert P^{\prime }{\Vert }_B=\sum |{\lambda }_j(y)|$. Also we formulate a geometric conjecture which implies that $\Vert P^{\prime }{\Vert }_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$. Keywords: regular simplex, ball, linear interpolation, projector, norm

13 pages

References in corpus (4)

• Five-dimensional Perfect Simplices
• Interpolation by Linear Functions on an $n$-Dimensional Ball
• On Some Problems Related to a Simplex and a Ball
• Geometric Estimates in Interpolation by Linear Functions on a Euclidean Ball

Cited by in corpus (2)

• The Minimum Norm of a Projector under Linear Interpolation on a Euclidean Ball
• Optimal Lagrange Interpolation Projectors and Legendre Polynomials