Deformations of Unbounded Convex Bodies and Hypersurfaces
نویسنده
چکیده
We study the topology of the space ∂K of complete convex hypersurfaces of R which are homeomorphic to Rn−1. In particular, using Minkowski sums, we construct a deformation retraction of ∂K onto the Grassmannian space of hyperplanes. So every hypersurface in ∂K may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of ∂K consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.
منابع مشابه
Tangent Cones and Regularity of Real Hypersurfaces
We characterize C embedded hypersurfaces of R as the only locally closed sets with continuously varying flat tangent cones whose measuretheoretic-multiplicity is at most m < 3/2. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is C. In the real analytic case the same conclusion holds under the weakened hypot...
متن کاملA Minkowski-style theorem for focal functions of compact convex reflectors
This paper1 continues the study of a class of compact convex hypersurfaces in R n+1 , n ≥ 1, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In R3 reflectors arise naturally in geometrical optics and are used in design of light reflectors and reflector antennas. They are al...
متن کاملSurvey Paper Combinatorial problems on the illumination of convex bodies
This is a review of various problems and results on the illumination of convex bodies in the spirit of combinatorial geometry. The topics under review are: history of the Gohberg–Markus–Hadwiger problem on the minimum number of exterior sources illuminating a convex body, including the discussion of its equivalent forms like the minimum number of homothetic copies covering the body; generalizat...
متن کاملWeighted composition operators between growth spaces on circular and strictly convex domain
Let $Omega_X$ be a bounded, circular and strictly convex domain of a Banach space $X$ and $mathcal{H}(Omega_X)$ denote the space of all holomorphic functions defined on $Omega_X$. The growth space $mathcal{A}^omega(Omega_X)$ is the space of all $finmathcal{H}(Omega_X)$ for which $$|f(x)|leqslant C omega(r_{Omega_X}(x)),quad xin Omega_X,$$ for some constant $C>0$, whenever $r_{Omega_X}$ is the M...
متن کاملHedgehog theory via Euler calculus
Hedgehogs are (possibly singular and self-intersecting) hypersurfaces that describe Minkowski differences of convex bodies in R. They are the natural geometrical objects when one seeks to extend parts of the Brunn-Minkowski theory to a vector space which contains convex bodies. In terms of characteristic functions, Minkowski addition of convex bodies correspond to convolution with respect to th...
متن کامل