Best Approximations by Vector-Valued Monotone Increasing or Convex Functions

Let (E, || · ||) be an ordered Banach space over the real held. Let BV be the space of all E-valued functions on a compact interval [a, b] ⊂ R which are of bounded variation. When BV is endowed with a norm || f ||v = || f(a)|| + v(f), f ∈ BV, where v(f) is a total variation of f on [a, b], we are concerned with the existence and uniqueness of best approximations by monotone increasing or convex functions of BV.