  Set systems of finite VC dimension are frequently used in applications
relating to machine learning theory and statistics. Two simple types of VC
classes which have been widely studied are the maximum classes (those which are
extremal with respect to Sauer's lemma) and so-called Dudley classes, which
arise as sets of positivity for linearly parameterized functions. These two
types of VC class were related by Floyd, who gave sufficient conditions for
when a Dudley class is maximum. It is widely known that Floyd's condition
applies to positive Euclidean half-spaces and certain other classes, such as
sets of positivity for univariate polynomials.
  In this paper we show that Floyd's lemma applies to a wider class of linearly
parameterized functions than has been formally recognized to date. In
particular we show that, modulo some minor technicalities, the sets of
positivity for any linear combination of real analytic functions is maximum on
points in general position. This includes sets of positivity for multivariate
polynomials as a special case.
