Nonlinear Images of Sets. I: Strong and Weak Convexity

Abstract

<jats:p> Developing the Polyak convexity principle, we show that strong convexity of a set in a Hilbert space is preserved under a <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C^{1,1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C^{1,1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> 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. </jats:p>