Lipschitz continuations of linearly bounded functions

V. A. Milman

Sbornik. Mathematics, Tome 189 (1998) no. 8, pp. 1179-1203

Résumé

The problem of the continuation of a real-valued function from a subset $Y$ of a metric space $(X,d)$ to the whole of the space is considered. A well-known result of McShane enables one to extend a uniformly continuous function preserving its modulus of continuity. However, some natural questions remain unanswered in the process. A new scheme for the extension of a broad class of functions, including bounded and Lipschitz functions, is proposed. Several properties of these extensions, useful in applications, are proved. They include the preservation of constraints on the increments of a function defined in terms of quasiconcave majorants. This result enables one to refine and generalize well-known results on the problem of the traces of functions with bounded gradient. The extension in question is used in two problems on function approximation. In particular, a direct proof of the density of the class $\operatorname {Lip}(X)$ in $\operatorname {lip}(X,\omega )$ is given.