Property A and best L1-Approximation

An important property within the linear theory of best L1-approximation, the so-called Property A, is discussed. Indeed, this property characterizes all finite-dimensional L1(μ)- unicity subspaces of continuous functions with respect to measures μ of a big class. Moreover, some extensions are considered: Property Ak, characterizing those subspaces where the sets of best L1(μ)-approximations have at most dimension k, and Property Aper characterizing L1(μ)-unicity subspaces of continuous periodic functions.