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An Upper Bound for Volumes of Convex Bodies

Published online by Cambridge University Press:  09 April 2009

William J. Firey
William J. Firey
Affiliation:
University of OtagoDunedin, N.Z. and Oregon State UniversityCorvallis Ore, U.S.A.
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Consider a non-degenerate convex body K in a Euclidean (n + 1)-dimensional space of points (x, z) = (x1,…, xn, z) where n ≧2. Denote by μ the maximum length of segments in K which are parallel to the z-axis, and let Aj, signify the area (two dimensional volume) of the orthogonal projection of K onto the linear subspace spanned by the z- and xj,-axes. We shall prove that the volume V(K) of K satisfies After this, some applications of (1) are discussed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 3-4 , May 1969 , pp. 503 - 510
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Bonnesen, T. and Fenchel, W., Theorie der honvexen Körper, (Berlin, 1934).Google Scholar
[2]Firey, W., ‘Lower bounds for volumes of convex bodies’, Archiv der Math. 16 (1965), 69–74.CrossRefGoogle Scholar
[3]Hardy, G., Littlewood, J. and Pólya, G., Inequalities, (cambridge, 1934).Google Scholar