Right Convexity

Abstract

<jats:p> A convex set is <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{F}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">F</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex if every pair of points in the set lie in a right triangle included in the set. We characterize <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{F}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">F</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex sets, find some classes of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{F}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">F</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex sets, investigate <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{F}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">F</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convexity for cones and cylinders, and find out that most convex bodies are <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{F}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">F</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex. On our way, we also describe the curvature at the endpoints of diameters of most convex bodies </jats:p>