On Constant Width Sets in Hilbert Spaces and Around

Abstract

<jats:p> We present some results concerning the family <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{CW}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">C</mml:mi> <mml:mi mathvariant="script">W</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of sets with constant width and the family <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{DM}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">D</mml:mi> <mml:mi mathvariant="script">M</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of diametrically maximal (or diametrically complete) sets. In particular we prove (Theorem 2) that <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{CW} = \mathcal{DM}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">C</mml:mi> <mml:mi mathvariant="script">W</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mi mathvariant="script">D</mml:mi> <mml:mi mathvariant="script">M</mml:mi> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> in Hilbert spaces. </jats:p>