ABSTRACT
Assume K is a complete nonarchimedean nontrivially valued field, A is a subset of K, and f : A → K is an isometric map, or an isometry, meaning a (not necessarily onto) map for which ∀ x , y ∈ A , | f ( x ) − f ( y ) | = | x − y | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq276.tif"/> holds. It is well-known, [2, Theorem 6], that when K is spherically complete, the map f can always be extended isometrically at least up to the set of so-called strictly efficient points of K with respect to A, namely c K ( A ) : = { x ∈ K : ∀ y ∈ K , ∃ a ∈ A , d ( a , x ) < d ( a , y ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq277.tif"/> .
