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Convolutions, Transforms, and Convex Bodies
Author(s) -
Grinberg Eric,
Zhang Gaoyong
Publication year - 1999
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611599001653
Subject(s) - mathematics , intersection (aeronautics) , mixed volume , regular polygon , convolution (computer science) , unit sphere , surface (topology) , convex body , ellipsoid , pure mathematics , trigonometric functions , mathematical analysis , geometry , convex hull , machine learning , artificial neural network , physics , astronomy , computer science , engineering , aerospace engineering
The paper studies convex bodies and star bodies in R n by using Radon transforms on Grassmann manifolds, p ‐cosine transforms on the unit sphere, and convolutions on the rotation group of R n . It presents dual mixed volume characterizations of i ‐intersection bodies and L p ‐balls which are related to certain volume inequalities for cross sections of convex bodies. It considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties. Convolution techniques are used to derive formulas for mixed volumes, mixed surface measures, and p ‐cosine transforms. They are also used to prove characterizations of geometric functionals, such as surface area and dual quermassintegrals. 1991 Mathematics Subject Classification : 52A20, 52A40.