Neighbourhood Retractions of Nonconvex Sets in a Hilbert Space via Sublinear Functionals
Abstract
<jats:p> For a closed subset <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of a Hilbert space <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\left( H,\left\Vert \cdot \right\Vert \right)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo fence="true">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:mrow> <mml:mo fence="true">∥</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="true">∥</mml:mo> </mml:mrow> <mml:mo fence="true">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and for a sublinear functional <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\rho:H\rightarrow \mathbb{R}^{+}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>H</mml:mi> <mml:mo>→</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo lspace="0em" rspace="0em">+</mml:mo> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , which is equivalent to the norm <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\left\Vert \cdot \right\Vert</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo fence="true">∥</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="true">∥</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , we give conditions guaranteeing existence and uniqueness of the nearest points to <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> in the sense of the semidistance generated by <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\rho</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . This permits us to construct a continuous retraction onto <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> well defined in a neighbourhood <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{U}\supset C</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">U</mml:mi> <mml:mo>⊃</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . In particular, according to one of the conditions, <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{U}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">U</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> can be represented in terms of balance between the local strict convexity modulus of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\rho</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and the measure of nonconvexity of the set <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> at each point. </jats:p>