A class of functions whose sublevel sets are absolute retracts (Q924280)

The main result is the following theorem: Let \(E\) be a real Banach space and let \(X\subset E\) be a closed and convex set. Moreover let \(\phi :X\to {\mathcal R}\) be a lower semicontinuous convex function such that \(\phi (x_0) = 0\) and \[ \alpha = \inf _{x_0\neq x \in X}\frac{\phi (x)}{| | x - x_0| | } > 0 \] for some \(x_0\in X\). If \(J:X\to {\mathcal R}\) is a Lipschitzian function with constant \(L< \alpha \) then each nonempty sublevel set of \(\phi + J\) is an absolute retract.