scholarly journals Localized Gaussian width of $M$-convex hulls with applications to Lasso and convex aggregation

Bernoulli ◽  
2019 ◽  
Vol 25 (4A) ◽  
pp. 3016-3040
Author(s):  
Pierre C. Bellec
Keyword(s):  
Convex Hulls ◽  
Gaussian Width

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.

Convex hulls of objects bounded by algebraic curves

Algorithmica ◽  
1991 ◽  
Vol 6 (1-6) ◽  
pp. 533-553 ◽  
Author(s):  
Chanderjit Bajaj ◽  
Myung -Soo Kim
Keyword(s):  
Algebraic Curves ◽  
Convex Hulls

scholarly journals Computable representations for convex hulls of low-dimensional quadratic forms

2010 ◽  
Vol 124 (1-2) ◽  
pp. 33-43 ◽  
Author(s):  
Kurt M. Anstreicher ◽  
Samuel Burer
Keyword(s):  
Quadratic Forms ◽  
Convex Hulls ◽  
Low Dimensional

Convex hulls of finite sets of points in two and three dimensions

1977 ◽  
Vol 20 (2) ◽  
pp. 87-93 ◽  
Author(s):  
F. P. Preparata ◽  
S. J. Hong
Keyword(s):  
Three Dimensions ◽  
Convex Hulls ◽  
Finite Sets
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