Nonlinear Images of Sets. I: Strong and Weak Convexity

Developing the Polyak convexity principle, we show that strong convexity of a set in a Hilbert space is preserved under a $C^{1,1}$ covering mapping provided that the parameters of the mapping and the set are related properly. We also prove that weak convexity of a set is preserved under $C^{1,1}$ bi-Lipschitz mapping and obtain exact estimates of the convexity parameters of the images of the Cartesian product of two sets depending on the parameters of the sets and the mapping.