  In Micchelli's paper "Interpolation of scattered data: distance matrices and
conditionally positive functions", deep results were obtained concerning the
invertibility of matrices arising from radial basis function interpolation. In
particular, the Euclidean distance matrix was shown to be invertible for
distinct data. In this paper, we investigate the invertibility of distance
matrices generated by $p$-norms. In particular, we show that, for any $p\in (1,
2)$, and for distinct points $ x^1, ..., x^n \in {\cal R}^d $, where $n$ and
$d$ may be any positive integers, with the proviso that $ n \ge 2$, the matrix
$A \in {\cal R}^{n \times n} $ defined by $$ A_{ij} = \Vert x^i - x^j \Vert_p ,
\hbox{ for } 1 \le i, j \le n, $$ satisfies $$ (-1)^{n-1}\det A > 0 .$$ We also
show how to construct, for every $p > 2$, a configuration of distinct points in
some ${\cal R}^d$ giving a singular $p$-norm distance matrix. Thus radial basis
function interpolation using $p$-norms is uniquely determined by any distinct
data for $p \in (1,2]$, but not so for $p > 2$.

Numerical Analysis