  This paper is concerned with Lagrange interpolation by total degree
polynomials in moderate dimensions. In particular, we are interested in
characterising the optimal choice of points for the interpolation problem,
where we define the optimal interpolation points as those which minimise the
Lebesgue constant. We give a novel algorithm for numerically computing the
location of the optimal points, which is independent of the shape of the domain
and does not require computations with Vandermonde matrices. We perform a
numerical study of the growth of the minimal Lebesgue constant with respect to
the degree of the polynomials and the dimension, and report the lowest values
known as yet of the Lebesgue constant in the unit cube and the unit ball in up
to 10 dimensions.
