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arXiv:math.MG/0502345v116Feb2005 MSC 52A20, 52A38, 52B11, 52B55, 49Q20, 65D17, 68U07
 

Summary: arXiv:math.MG/0502345v116Feb2005
MSC 52A20, 52A38, 52B11, 52B55, 49Q20, 65D17, 68U07
Blaschke Addition and Convex Polyhedra
Victor Alexandrov1
, Natalia Kopteva, S. S. Kutateladze
Abstract. This is an extended version of a talk on October 4, 2004 at the
research seminar "Differential geometry and applications" (headed by Academician
A. T. Fomenko) at Moscow State University. The paper contains an overview of
available (but far from well-known) results about the Blaschke addition of convex
bodies, some new theorems on the monotonicity of the volume of convex bodies (in
particular, convex polyhedra with parallel faces) as well as description of a software
for visualization of polyhedra with prescribed outward normals and face areas.
1. The vector area of the surface of a polyhedron. In this article we take
the following fact as well-known: Let P be a compact convex polyhedron in Rm
(m 2), n1, . . . , nk be unit vectors of outward normals to its (m - 1)-dimensional
faces and let F1, . . . , Fk be (m - 1)-dimensional volumes of its (m - 1)-dimensional
faces. Then
k
j=1
Fjnj = 0. (1)

  

Source: Alexandrov, Victor - Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk

 

Collections: Mathematics