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Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Part of: Real functions: Miscellaneous topics Polytopes and polyhedra

Published online by Cambridge University Press:  18 February 2021

Joseph Malkoun Open the ORCID record for Joseph Malkoun [Opens in a new window]  and
Peter J. Olver Open the ORCID record for Peter J. Olver [Opens in a new window]
Joseph Malkoun
Affiliation:
Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaize, Zouk Mosbeh, P.O. Box 72, Zouk Mikael, Lebanon; E-mail: joseph.malkoun@ndu.edu.lb
Peter J. Olver
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA; E-mail: olver@umn.edu

Abstract

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Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ 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.

MSC classification

Primary: 52B55: Computational aspects related to convexity
Secondary: 26E25: Set-valued functions
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e13
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Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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