Spherical Projections and Centrally Symmetric Sets

✍ Scribed by Hermann Fallert; Paul Goodey; Wolfgang Weil

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 356 KB
• Volume: 129
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1657

✦ Synopsis

Centrally symmetric convex bodies in d-dimensional Euclidean space R d are related to various transforms of functions on the unit sphere S d&1 in R d . In this paper we will investigate how this relationship is affected by projections of the bodies onto lower dimensional subspaces. The results we obtain will also give information about central sections of certain starshaped sets.

The cosine transform T, in R d , is defined on the Montel space C e of even, infinitely differentiable functions on S d&1 by

where ( } , } ) is the usual scalar product in R d and * j denotes the j-dimensional spherical Lebesgue measure. We will denote by K 0 the class of all convex bodies (non-empty, compact, convex sets) which are centrally symmetric with respect to the origin. In it was shown that, corresponding article no. AI971657 301