Rotundity Properties, and Non-Extendability of Lipschitz Quasiconvex Functions

Let A be an open convex subset of a real normed linear space X. We prove that if the boundary of A contains a non-LUR point then there exists a Lipschitz quasiconvex function f from A to $\mathbb{R}$ not admitting any continuous quasiconvex extension to the whole X.