Abstract:
Let $S$ be a nondegenerate simplex in ${\mathbb R}^n$.
Denote by $\alpha(S)$ the minimal $\sigma>0$ such that
the unit cube $Q_n:=[0,1]^n$
is contained in a translate of $\sigma S$. In the case $\alpha(S)\ne 1$
the translate of $\alpha(S)S$ containing $Q_n$ is a homothetic copy of $S$
with the homothety center at some point
$x\in{\mathbb R}^n$.
We obtain the following computational formula for $x$.
Denote by $x^{(j)}$$(j=1,\ldots, n+1)$ the vertices of $S$.
Let
${\mathbf A}$ be the
matrix of order $n+1$ with the rows
consisting of the
coordinates of $x^{(j)};$ the last column of ${\mathbf A}$ consists
of 1's. Suppose that ${\mathbf A}^{-1}=(l_{ij}).$ Then the coordinates
of $x$ are the numbers
$$x_k=
\frac{\sum_{j=1}^{n+1}
\left(\sum_{i=1}^n \left|l_{ij}\right|\right)x^{(j)}_k
-1}
{\sum_{i=1}^n\sum_{j=1}^{n+1} |l_{ij}|- 2} \quad (k=1,\ldots,n).$$
Since $\alpha(S)\ne 1,$ the denominator from the right-hand part of this
equality is not equal to
zero.
Also we give the estimates
for norms of projections dealing with the linear interpolation of
continuous functions defined on $Q_n$.
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