Right Triple Convex Completion

Abstract

<jats:p> A set M in <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{R}^d</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>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is rt-convex if every pair of its points is included in a 3-point subset {x,y,z} of M satisfying <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\angle xyz = \pi/2</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">∠</mml:mi> <mml:mi>x</mml:mi> <mml:mi>y</mml:mi> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> <mml:mi mathvariant="normal">/</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . We find here for various families of sets the minimal number of points necessary to add to the sets in order to render them rt-convex </jats:p>