Several generalizations of the ordinary variation diminishing property are derived in the sense that, given an \(m \times n\) totally positive matrix \(T\) and an \(n \times r\) matrix \(A\) satisfying some additional conditions, then the number of changes of sign in the consecutive \(r \times r\) minors of \(TA\) is bounded by the number of changes of sign in the consecutive \(r \times r\) minors of \(A\). This property is used to show shape preserving properties of curves generated by totally positive bases and, in particular, of \(B\)-spline curves.