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Stability in the Aleksandrov-Fenchel-Jessen Theorem

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

R. Schneider
R. Schneider
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universitt, D-7800 Freiburg i. Br., FRG.
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Extract

The theorem of Aleksandrov-Fenchel-Jessen states that two convex bodies in n-dimensional Euclidean space En which, for some p l, , n - l 007D;, have equal area measures of order p (see Section 2 for a definition) differ only by a translation. Two independent proofs were given by Aleksandrov 1 and by Fenchel and Jessen 18 see also Busemann 5 (p. 70) and LeichtweiG 25 (p. 319), 26. If the boundaries of the two bodies are sufficiently smooth and of everywhere positive curvatures, then the assumption of the theorem is equivalent to saying that at points with parallel outer normals the p-th elementary symmetric functions of the principal radii of curvature of both boundary hypersurfaces are the same. For this case, Chern 6 gave a uniqueness proof by means of an integral formula.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)

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Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 50 - 59
Copyright
Copyright University College London 1989

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