If the Basis Functions are Somewhere Negative

The condition on a non-negative basis function is essentially a condition on the Zoo norm of the matrix of basis functions, introduced in the last chapter, maxt Ei b(t — i) 1 which, because Sib(t — i) = 1, cannot be less than 1. 

Consider the argument above, which said that the convex hull only worked if Z, the Zoo norm, was less than or equal to 1. This same argument can still be applied to construct an enclosure when Z 1. 

We merely have to scale up the convex hull by a factor of Z. To be more precise, each band corresponding to a face orientation needs to be scaled up about its centroid by a factor of Z. 

Thus enclosures can indeed be found for schemes which are not positive ones. 

The Zoo norm graphed over one span is symmetric, and it looks as though the maximum is at the central point of a span. In fact it is not quite. abs(f(x)) is itself a function, with V-shaped discontinuities of slope at the roots of f(x). 

---