On Abstract Convexity with Respect to a Certain Quadratic Coupling Function: Subsets
Abstract
<jats:p> We investigate abstract convexity generated by the coupling function <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varphi(x,A)=|A[x]^2|,\quad x\in \mathbb{R}^n,\ A\in \mathcal{L}^2_{sym}(\mathbb{R}^n,\mathbb{R}).</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∣</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mi mathvariant="normal">∣</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:mspace width="1em"/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo separator="true">,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>A</mml:mi> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi mathvariant="script">L</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mi>y</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo separator="true">,</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="normal">.</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> We study the <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varphi</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convexity of four classes of sets: subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{R}^n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{L}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , epigraphic subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{R}^n\times \mathbb{R}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and epigraphic subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{L}\times\mathbb{R}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">L</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . We show that all families consist of closed, star-shaped sets and subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{L}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{L}\times\mathbb{R}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">L</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> are convex. For epigraphic subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X\times\mathbb{R}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , we derive a condition partially characterizing them in analogy with classical convexity, where line segments are replaced by parabolas. In the one-dimensional case we obtain a complete characterization </jats:p>