Total positivity and optimal bases

In a finite dimensional space which has a totally positive basis there exist special bases, called B-bases, such that they generate all totally positive bases by means of totally positive matrices. B-bases are optimal totally positive bases in several senses. From the geometrical point of view, B-bases correspond to the bases with optimal shape preserving properties. From a numerical point of view B-bases are least supported and least conditioned bases among all totally positive bases of space. In order to deal with these questions we introduce a partial order in the set of nonnegative bases: if (v₀,..., v _n ) = (u₀,..., u _n )H for a nonnegative matrix H, we say that (u₀,..., u _n ) ≤ (v₀,..., v _n ) . We shall show that B-bases are minimal for this partial order.