We prove the following proposition. Under linear interpolation on a Euclidean $n$-dimensional ball $B$, an interpolation projector whose nodes coincide with the vertices of a regular simplex inscribed into the boundary sphere has the minimum $C$-norm. This minimum norm $\theta_n(B)$ is equal to $\max\{ \psi(a_n),\psi(a_n+~1)\}$, where $\psi(t)=\dfrac{ 2\sqrt{ n}}{ n+1}\Bigl(t(n+1-t)\Bigr)^{ 1/2}+ \left|1-\dfrac{ 2t}{ n+1}\right|$, $0\leq t\leq n+1$, and $a_n=\left\lfloor\dfrac{ n+1}{ 2}-\dfrac{ \sqrt{ n+1}}{ 2}\right\rfloor$. For any $n$, $\sqrt{ n}\leq \theta_n(B)\leq \sqrt{ n+1}.$ Moreover, $\theta_n(B)$ $=$ $\sqrt{ n}$ only for $n=1$ and $\theta_n(B)=\sqrt{ n+1}$ if and only if $\sqrt{ n+1}$ is an integer.
Categories: math.MG