On extension of uniformly continuous quasiconvex functions

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 ⊂ 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.