  In the classical best approximation pair (BAP) problem, one is given two
nonempty, closed, convex and disjoint subsets in a finite- or an
infinite-dimensional Hilbert space, and the goal is to find a pair of points,
each from each subset, which realizes the distance between the subsets. This
problem, which has a long history, has found applications in science and
technology. We discuss the problem in more general normed spaces and with
possibly non-convex subsets, and focus our attention on the issues of
uniqueness and existence of the solution to the problem. To the best of our
knowledge these fundamental issues have not received much attention. In
particular, we present several sufficient geometric conditions for the (at
most) uniqueness of a BAP relative to these subsets. These conditions are
related to the structure of the boundaries of the subsets, their relative
orientation, and the structure of the unit sphere of the space. In addition, we
present many sufficient conditions for the existence of a BAP, possibly without
convexity . Our results allow us to significantly extend the horizon of the
recent alternating simultaneous Halpern-Lions-Wittmann-Bauschke (A-S-HLWB)
algorithm [Censor, Mansour and Reem, The alternating simultaneous
Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation
pair for two disjoint intersections of convex sets, arXiv:2304.09600 (2023)]
for solving the BAP problem.
