  As an analogy of best separable approximation (BSA) in the framework of
entanglement theory, here we concentrate on the notion of best incoherent
approximation, with application to characterizing and quantifying quantum
coherence. From both analytical and numerical perspectives, we have
demonstrated that the weight-based coherence measure displays some unusual
properties, in sharp contrast to other popular coherence quantifiers. First, by
deriving a closed formula for qubit states, we have showed the weight-based
coherence measure exhibits a rich (geometrical) structure even in this simplest
case. Second, we have identified the existence of mixed maximally coherent
states (MMCS) with respect to this coherence measure and discussed the
characteristic feature of MMCS in high-dimensional Hilbert spaces. Especially,
we present several important families of MMCS by gaining insights from the
numerical simulations. Moreover, it is pointed out that some considerations in
this work can be generalized to general convex resource theories and a
numerical method of improving the computational efficiency for finding the BSA
is also discussed.
