Hilbert points in Hilbert space-valued $L^p$ spaces

Abstract

Let H be a Hilbert space and (Ω,F ,µ) a probability space. A Hilbert point in L p (Ω;H) is a nontrivial function ϕ such that ∥ϕ∥p ≤ ∥ϕ+f ∥p whenever ⟨f ,ϕ⟩ = 0. We demonstrate that ϕ is a Hilbert point in L p (Ω;H) for some p , 2 if and only if ∥ϕ(ω)∥H assumes only the two values 0 and C > 0. We also obtain a geometric description of when a sum of independent Rademacher variables is a Hilbert point.