Linear combinations of convex hypersurfaces in non-Euclidean geometry
type: Item
instance of: scholarly article
title: Linear combinations of convex hypersurfaces in non-Euclidean geometry
zbMATH Open document ID: 1242.52013
author: Kurt Leichtweiß
published in: Beiträge zur Algebra und Geometrie
publication date: 2012-02-29T00:00:00Z
review text: The author defines the Minkowski sum of two arbitrary strongly convex bodies in the framework provided by the Riemannian geometry (\(n\)-dimensional spherical geometry). The approach of the same problem in the Weierstrass geometry (\(n\)-dimensional hyperbolic geometry) leads to satisfactory results in the restricted case of distance convex bodies having \(\bar{0}=(1,0,\dots,0)\) in its interior. The resulted convex bodies of this operation are described in terms of a support function, which is implicitly expressed in both cases. The sum operation is extended to convex combinations and, as a consequence, the arithmetic mean of two convex sets is discussed in hyperbolic geometry.
reviewed by: Gabriela Cristescu
Mathematics Subject Classification ID: 52A55
zbMATH DE Number: 6010036
zbMATH Keywords: linear combination of convex bodies
zbMATH Keywords: radial map of the spherical model
zbMATH Keywords: radial map of the Weierstrass model
zbMATH Keywords: distance convex bodies in hyperbolic space
zbMATH Keywords: quasi support functions in spherical space
MaRDI profile type: Publication
full work available at URL: s13366-011-0080-4
OpenAlex ID: W1979419343
cites work: Asymptotic behaviour of \(\lambda\)-convex sets in the hyperbolic plane
cites work: Linear combinations of hypersurfaces in hyperbolic space
cites work: Support function and hyperbolic plane
cites work: Curves of constant width in the non-Euclidean geometry
cites work: Polar curves in the non-Euclidean geometry
cites work: Q5287551
cites work: On algebraic structures related to Beltrami-Klein model of hyperbolic geometry
cites work: Konvexität in riemannschen Mannigfaltigkeiten
DOI: 10.1007/S13366-011-0080-4
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description: scientific article; zbMATH DE number 6010036