On extension of uniformly continuous quasiconvex functions

Abstract:

We show that each uniformly continuous quasiconvex function defined on a subspace of a normed space $X$ admits a uniformly continuous quasiconvex extension to the whole $X$ with the same “invertible modulus of continuity”. This implies an analogous extension result for Lipschitz quasiconvex functions, preserving the Lipschitz constant.

We also show that each uniformly continuous quasiconvex function defined on a uniformly convex set $A\subset X$ admits a uniformly continuous quasiconvex extension to the whole $X$. However, our extension need not preserve moduli of continuity in this case, and a Lipschitz quasiconvex function on $A$ may admit no Lipschitz quasiconvex extension to $X$ at all.