Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Joseph Malkoun, Peter J. Olver

Research output: Contribution to journalArticlepeer-review

Abstract

Given n distinct points in, let K denote their convex hull, which we assume to be d-dimensional, and its-dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps which, for 0$]]>, are defined on the-dimensional sphere, and whose images are codimension submanifolds contained in the interior of K. Moreover, as the parameter goes to, the images converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

Original languageEnglish (US)
Article numbere13
JournalForum of Mathematics, Sigma
DOIs
StateAccepted/In press - 2021

Bibliographical note

Publisher Copyright:
© The Author(s), 2021. Published by Cambridge University Press.

Keywords

  • 2020 Mathematics Subject Classification
  • 26E25
  • 52B55

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