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On hyperbolic virtual polytopes and hyperbolic fans

Gaiane Panina (2006)

Open Mathematics

Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R 1 ≤ C ≤ R 2 then K is a ball. (R 1 and R 2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic...

Sets invariant under projections onto two dimensional subspaces

Simon Fitzpatrick, Bruce Calvert (1991)

Commentationes Mathematicae Universitatis Carolinae

The Blaschke–Kakutani result characterizes inner product spaces E , among normed spaces of dimension at least 3, by the property that for every 2 dimensional subspace F there is a norm 1 linear projection onto F . In this paper, we determine which closed neighborhoods B of zero in a real locally convex space E of dimension at least 3 have the property that for every 2 dimensional subspace F there is a continuous linear projection P onto F with P ( B ) B .

Testing of convex polyhedron visibility by means of graphs

Jozef Zámožík, Viera Zat'ková (1980)

Aplikace matematiky

This paper follows the article by V. Medek which solves the problem of finding the boundary of a convex polyhedron in both parallel and central projections. The aim is to give a method which yields a simple algorithm for the automation of an arbitrary graphic projection of a convex polyhedron. Section 1 of this paper recalls some necessary concepts from the graph theory. In Section 2 graphs are applied to determine visibility of a convex polyhedron.

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