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Approximation of convex surfaces by algebraic surfaces

Published online by Cambridge University Press:  26 February 2010

P. C. Hammer
P. C. Hammer
Affiliation:
University of California, San Diego, California.
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Extract

Minkowski [1] first proved that the surface of a convex body in E3 can be approximated by a level surface of a convex analytic function. His proof is strikingly simple. His proof is also presented for En by Bonnesen-Fenchel [2, pp. 10–12[. We here prove that the same kind of result is achieved using level surfaces of convex non-negative polynomials. We give two types of approximation, one based on finite sums as Minkowski did, and the other using integration. Since these approximations may be used for other applications we also extend them and give special formulae when the surface is centrally symmetric.

Type
Research Article
Information
Mathematika , Volume 10 , Issue 1 , June 1963 , pp. 64 - 71
Copyright
Copyright © University College London 1963

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References

1. Minkowski, H., “Volumen und Oberfläche”, Math. Annalen, 57 (1903), 447–495.CrossRefGoogle Scholar
2. Bonnosen, T. and Fenchel, W., Theorie der Convexen Korper (Julius Springer, Berlin 1934; Chelsea, New York 1948).Google Scholar