TY - JOUR

T1 - Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

AU - Malkoun, Joseph

AU - Olver, Peter J.

N1 - Publisher Copyright:
© The Author(s), 2021. Published by Cambridge University Press.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - 2020 Mathematics Subject Classification

KW - 26E25

KW - 52B55

UR - http://www.scopus.com/inward/record.url?scp=85100884675&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85100884675&partnerID=8YFLogxK

U2 - 10.1017/fms.2021.10

DO - 10.1017/fms.2021.10

M3 - Article

AN - SCOPUS:85100884675

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

SN - 2050-5094

M1 - e13

ER -