On the Infimum of a Quasiconvex Function over an Intersection. Application to the Distance Function

We give sufficient conditions for the infimum of a quasiconvex function

over the intersection

\bigcap_{i\in I}R_{i} ⋂ i ∈ I R i

to agree with the supremum of the infima of

f f

over the

R_{i} R i

's. We apply these results to the distance function in a normed space.