### On Properties of a Regular Simplex Inscribed into a Ball

Mikhail Viktorovich Nevskii
2021 Modelirovanie i Analiz Informacionnyh Sistem
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 $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, more » ... .The norm of$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\$.