On some Class of Polytopes in an Idempotent, Symmetrical and Non-Associative Convex Structure
Abstract
<jats:p> <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convexity was defined by the author and C. D. Horvath [ <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convexity, Optimization 53(2) (2004) 103–127] as a suitable Painlevé-Kuratowski limit of linear convexities. Recently, an alternative algebraic formulation over the whole Euclidean vector space was proposed by the author in further articles [Some remarks on an idempotent and non-associative convex structure, J. Convex Analysis 22 (2015) 259–289; and Separation properties in some idempotent and symmetrical convex structure, J. Convex Analysis 24 (2017) 1143–1168]. In this paper the structure of the polytopes arising in such a context is analyzed. It is first established that this notion of convexity has the interval and decompositions properties but fails the recursive property. Among the key results of the paper, it is proven that a polytope defined over <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> is the Painlevé-Kuratowski limit of a suitable sequence of generalized convex polytopes. Some analogues to the Helly, Radon and Caratheodory theorems are derived in the limit. A separation theorem for <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -polytopes is stated extending an earlier result recently established by the author [Separation properties in some idempotent and symmetrical convex structure, J. Convex Analysis 24 (2017) 1143–1168] for fixed <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex sets. Finally, considering the Kuratowski-Painlevé limit of a suitable sequence of linear halfspaces the external representation of a <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -polytope is established over <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> . It follows that the interior of a <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -polytope defined over <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> satisfies the convexity criterion proposed by the author previously [Some remarks on an idempotent and non-associative convex structure, J. Convex Analysis 22 (2015) 259–289]. </jats:p>