Abstract

Given a compact and H-convex subset KK of the Heisenberg group H\H, with the origin ee in its interior, we are interested in finding a homogeneous H-convex function ff such that f(e)=0f(e)=0 and f∣∂K=1f\bigl|_{\partial K}=1; we will call this function ff the H\H-cone-function of vertex ee and base ∂K\partial K. While the equivalent version of this problem in the Euclidean framework has an easy solution, in our context this investigation turns out to be quite entangled, and the problem can be unsolvable. The approach we follow makes use of an extension of the notion of convex family introduced by Fenchel. We provide the precise, even if awkward, condition required to KK so that ∂K\partial K is the base of an H\H-cone-function of vertex e.e. Via a suitable employment of this condition, we prove two interesting binding constraints on the shape of the set K,K, together with several examples.

Author information

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

Andrea Calogero

Dip. di Matematica e Applicazioni, Università di Milano-Bicocca, 20125 Milano, Italy

andrea.calogero@unimib.it

Rita Pini

Dip. di Matematica e Applicazioni, Università di Milano-Bicocca, 20125 Milano, Italy

rita.pini@unimib.it

Keywords

  • Heisenberg group
  • H-convexity
  • convex families
  • cone-functions

Mathematics Subject Classification

26B25; 53C17, 22E30, 22E25

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Suggested citation

A. Calogero, R. Pini. “On the H-Cone-Functions for H-Convex Sets.” Journal of Convex Analysis 26 (2019), No. 3, 967–989. https://doi.org/10.68381/jca26052

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.