On Properties of Different Notions of Centers for Convex Cones

The points on the revolution axis of a circular cone are somewhat special: they are the “most interior” elements of the cone. This paper addresses the issue of formalizing the concept of center for a convex cone that is not circular. Four distinct proposals are studied in detail: the incenter, the circumcenter, the inner center, and the outer center. The discussion takes place in the context of a reflexive Banach space.

[1]  Robert M. Freund,et al.  A geometric analysis of Renegar’s condition number, and its interplay with conic curvature , 2007, Math. Program..

[2]  Rainald Löhner,et al.  On the 'most normal' normal , 2007 .

[3]  Robert E. Megginson.  An Introduction to Banach Space Theory , 1998 .

[4]  Per Enflo,et al.  Banach spaces which can be given an equivalent uniformly convex norm , 1972 .

[5]  Jonathan M. Borwein,et al.  Differentiability of cone-monotone functions on separable Banach space , 2003 .

[6]  Gerald Beer,et al.  Topologies on Closed and Closed Convex Sets , 1993 .

[7]  C. Shannon.  Probability of error for optimal codes in a Gaussian channel , 1959 .

[8]  Roger J.-B. Wets,et al.  Continuity of some convex-cone-valued mappings , 1967 .

[9]  René Henrion,et al.  Inradius and Circumradius of Various Convex Cones Arising in Applications , 2010 .

[10]  D. Zhuang.  Bases of convex cones and Borwein's proper efficiency , 1991 .

[11]  R. Freund,et al.  On the Primal-Dual Geometry of Level Sets in Linear and Conic Optimization , 2001 .

[12]  Alfredo N. Iusem,et al.  Axiomatization of the index of pointedness for closed convex cones , 2005 .

[13]  Robert M. Freund,et al.  A New Condition Measure, Preconditioners, and Relations Between Different Measures of Conditioning for Conic Linear Systems , 2002, SIAM J. Optim..

[14]  Jean-Pierre Aubin.  Mutational and Morphological Analysis , 2012 .

[15]  Alfredo N. Iusem,et al.  Normality and modulability indices. Part II Convex cones in hilbert spaces , 2008 .

[16]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[17]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[18]  A. Seeger,et al.  Normality and modulability indices. Part II , 2008 .

[19]  A. Seeger,et al.  Normality and modulability indices . Part I : Convex cones in normed spaces , 2007 .

[20]  R. Holmes.  Geometric Functional Analysis and Its Applications , 1975 .

[21]  Robert M. Freund,et al.  On the Primal-Dual Geometry of Level Sets in Linear and Conic Optimization , 2001, SIAM J. Optim..

[22]  Alfredo N. Iusem,et al.  Searching for critical angles in a convex cone , 2009, Math. Program..

[23]  Robert M. Freund,et al.  Condition-Based Complexity of Convex Optimization in Conic Linear Form via the Ellipsoid Algorithm , 1999, SIAM J. Optim..

[24]  Jean-Louis Goffin,et al.  The Relaxation Method for Solving Systems of Linear Inequalities , 1980, Math. Oper. Res..

[25]  Kim C. Border,et al.  Infinite dimensional analysis , 1994 .

[26]  Z. Han,et al.  Remarks on the angle property and solid cones , 1994 .

[27]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[28]  Johannes Jahn,et al.  Vector optimization - theory, applications, and extensions , 2004 .