$U$-embedded subsets of normed linear spaces

A subset $S$ of a metric space $X$ is $U$-embedded in $X$ if every uniformly continuous function $f:S \to R$ extends to a uniformly continuous function $F:X \to R$. Thus $U$-embedding is the uniform analogue of $C$-embedding. The Tietze extension theorem tells us exactly which subsets of metric spaces are $C$-embedded. The uniform analogue would tell us exactly which subsets of metric spaces are $U$-embedded. In this paper, a characterization of $U$-embedded subsets of the Euclidean plane (or any normed linear space) is given.