Abstract

The paper studies convex radiant sets (i.e. containing the origin) of a linear normed space XX and their representation by means of a gauge. By gauge of a convex radiant set C⊆XC\subseteq X we mean a sublinear function p:X→Rˉp:X\to\R such that C=[p≤1]C=[p\leq 1]. Besides the most important instance, namely the Minkowski gauge μC(x)=inf⁡{λ>0: x∈λC}\mu_C(x)=\inf\{\lambda >0: \,x\in\lambda C\}, the set CC may have other gauges, which are necessarily lower than μC\mu_C. We characterize the class of convex radiant sets which admit a gauge different from μC\mu_C in two different way: they are contained in a translate of their recession cone or, equivalently, they are costarshaped, that is complement of a starshaped set. We prove that the family of all sublinear gauges of a convex radiant set admits a least element and characterize its support set in terms of polar sets. The key concept for this study is the outer kernel of CC, that is the kernel (in the sense of Starshaped Analysis) of the complement of CC. We also devote some attention to the relation between costarshaped and hyperbolic convex sets.

Author information

Contact details are reproduced from the original publication and may be historical.

Alberto Zaffaroni

Dip. di Economia, Università di Modena e Reggio Emilia, Viale Berengario 51, 41121 Modena, Italy

alberto.zaffaroni@unimore.it

Keywords

  • Convex sets
  • Minkowski gauge
  • sublinear gauge
  • radiant sets
  • costarshaped sets
  • kernel
  • outer kernel
  • polar set
  • reverse polar
  • hyperbolic convex sets

Mathematics Subject Classification

52A07, 46A55, 46B20

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Suggested citation

A. Zaffaroni. “Convex Radiant Costarshaped Sets and the Least Sublinear Gauge.” Journal of Convex Analysis 20 (2013), No. 2, 307–328. https://doi.org/10.68381/jca20020

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.