Hyperspaces of smooth convex bodies up to congruence

2017/05/03 by Igor Belegradek, Belegradek, Igor
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #General Topology (math.GN) #math.DG #math.GN

paper · pdf · doi:10.48550/arxiv.1705.01220

24 pages, editorial changes and a new result added in Theorem 1.2(7)

arxiv created 2017/06/07 · arxiv updated 2017/06/08

Abstract

We determine the homeomorphism type of the hyperspace of positively curved C^∞ convex bodies in \mathbb Rⁿ, and derive various properties of its quotient by the group of Euclidean isometries. We make a systematic study of hyperspaces of convex bodies that are at least C¹. We show how to destroy the symmetry of a family of convex bodies, and prove that this cannot be done modulo congruence.

Related

• Hyperspaces of convex bodies of constant width
• Hyperbolic geometry of shapes of convex bodies
• Deformations of unbounded convex bodies and hypersurfaces
• Expensive Homeomorphism of Convex Bodies
• Symmetry of convex sets and its applications to the extremal ellipsoids of convex bodies