Canonical models for a class of polynomially convex hulls

1997 ◽  
Vol 308 (1) ◽  
pp. 47-63 ◽  
Author(s):  
Zbigniew Slodkowski
Keyword(s):  
Convex Hulls ◽  
Canonical Models ◽  
Polynomially Convex

scholarly journals Polynomially convex hulls of families of arcs

2004 ◽  
Vol 165 (1) ◽  
pp. 1-17
Author(s):  
Zbigniew Słodkowski
Keyword(s):  
Convex Hulls ◽  
Polynomially Convex

scholarly journals Polynomially convex hulls of singular real manifolds

2015 ◽  
Vol 368 (4) ◽  
pp. 2469-2496 ◽  
Author(s):  
Rasul Shafikov ◽  
Alexandre Sukhov
Keyword(s):  
Convex Hulls ◽  
Polynomially Convex

scholarly journals Polynomially convex hulls of graphs on the sphere

1999 ◽  
Vol 127 (9) ◽  
pp. 2697-2702
Author(s):  
Toshiya Jimbo ◽  
Akira Sakai
Keyword(s):  
Convex Hulls ◽  
Polynomially Convex

scholarly journals Polynomially convex hulls and analyticity

1982 ◽  
Vol 20 (1-2) ◽  
pp. 129-135 ◽  
Author(s):  
John Wermer
Keyword(s):  
Convex Hulls ◽  
Polynomially Convex

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract 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.

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