THE GEOMETRY OF TWO-VALUED SUBSETS OF Lp-SPACES

Let M(Omega, mu) denote the algebra of all scalar-valued measurable functions on a measure space (Omega, mu). Let B subset of M(Omega, mu) be a set of finitely supported measurable functions such that the essential range of each f is an element of B is a subset of {0, 1}. The main result of this paper shows that for any p is an element of (0, infinity), B has strict p-negative type when viewed as a metric subspace of L-p(Omega, mu) if and only if B is an affinely independent subset of M(Omega, mu) (when M(Omega, mu) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L-p(Omega, mu) has strict p-negative type. For instance, for each p is an element of (0, infinity), the system of Walsh functions in L-p[0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p is an element of (2, infinity), subsets of L-p(Omega, mu) that have p-negative type but not q-negative type for any q > p. Such sets preclude the existence of certain types of isometry into L-p-spaces. (C) 2017 Mathematical Institute Slovak Academy of Sciences