The Lipschitz constants of the metric $\varepsilon$-projection operator in spaces with given modules of convexity and smoothness

We obtain upper estimates for the Lipschitz constants of the metric $\varepsilon$-projection operator $P$ in terms of the modules of convexity and smoothness of the space when the following three parameters are varied: the approximee $x$, the convex approximating set $M$, and the accuracy of approximation $\varepsilon>0$. These estimates are unimprovable in the class of all normed linear spaces. We use them to obtain new stability evaluations for continuous selectors of the operator $P$.