  Totally positive matrices are related with the shape preserving
representations of a space of functions. The normalized B-basis of the space
has optimal shape preserving properties. B-splines and rational Bernstein bases
are examples of normalized B-bases. Some results on the optimal conditioning
and on extremal properties of the minimal eigenvalue and singular value of the
collocation matrices of normalized B-bases are proved. Numerical examples
confirm the theoretical results and answer related questions.
