Given two nonempty and disjoint intersections of closed and convex subsets,

we look for a best approximation pair relative to them, i.e., a pair of points,

one in each intersection, attaining the minimum distance between the disjoint intersections.

We propose an iterative process based on projections onto the subsets

which generate the intersections. The process is inspired by the Halpern-Lions-

Wittmann-Bauschke algorithm and the classical alternating process of Cheney and

Goldstein, and its advantage is that there is no need to project onto the intersections

themselves, a task which can be rather demanding. We prove that under certain

conditions the two interlaced subsequences converge to a best approximation pair.

These conditions hold, in particular, when the space is Euclidean and the subsets

which generate the intersections are compact and strictly convex. Our result extends

the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of

points for two polyhedra", Computational Optimization and Applications 71 (2018),

509-523] which considered the case of finite-dimensional polyhedra.

## Citation

Preprint, April 13, 2023.