Conjugacy for closed convex sets
type: Item
instance of: scholarly article
title: Conjugacy for closed convex sets
zbMATH Open document ID: 1074.52002
author: Daniel A. Jaume
author: Rubén Puente
published in: Beiträge zur Algebra und Geometrie
publication date: 2005-04-06T00:00:00Z
full work available at URL: https://eudml.org/doc/226726
full work available at URL: 9.html
review text: The authors generalize the definition of conjugate faces of a convex body. For a face \(F\) of a closed convex set \(C\) containing the origin they propose the following definition of the conjugate face \(F^{\triangle}\). Namely, \(F^{\triangle} = \{ y \in C^o \;| \;\forall_{x \in F} \;\langle x,y \rangle = -1 \}\) provided this set is non-empty, and otherwise \(F^{\triangle} = \{ y \in 0^+(C^o) \;| \;\forall_{x \in F} \;\langle x,y \rangle = -1 \}\). Here \(C^o\) means the polar set of \(C\) and \(0^+(C)\) denotes the recession cone of \(C\). A number of properties of the introduced conjugate faces is shown. For instance: if \(F\) is a face of \(C\) and if \(F^{\triangle} \not = \emptyset\), then \(F^{\triangle}\) is an exposed face of \(C^o\).
reviewed by: Marek Lassak
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 15A39
zbMATH DE Number: 2153152
zbMATH Keywords: convex set
zbMATH Keywords: polar set
zbMATH Keywords: conjugate face
zbMATH Keywords: convex cone
zbMATH Keywords: curvature index
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2153152