An observation on the Gram matrices of systems of uniformly bounded functions and a problem of Olevskii

In what follows ⟨ · , · ⟩ and | · | denote the scalar product and Euclidean norm in R , SN−1 = {x ∈ R : |x| = 1}, and μN−1 is the normalized Lebesgue measure on SN−1, N = 2, 3, . . . . For an N × N matrix G, we let ∥G∥op denote the norm of G as an operator in (R , | · |). We also use the following notation: ( · , · ) is the inner product in the function space L and ∥ · ∥∞ is the norm in L∞(0, 1). Given a system of vectors Z = {zj}j=1 ⊂ R , consider the Gram matrix GZ = {⟨zj , zk⟩}, 1 ⩽ j, k ⩽ N . The problem of finding a system of functions F = {fj}j=1 ⊂ L∞(0, 1) with uniform norms as small as possible and such that