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Note on a conjecture of L. A. Santaló

Published online by Cambridge University Press:  26 February 2010

H. G. Eggleston
H. G. Eggleston
Affiliation:
Bedford College, Regent's Park, London, N.W.1.
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Let be a plane bounded convex set whose width in the direction φ is . It has been shown by L. A. Santaló that

is invariant under unimodular affine transformations. Santaló [1] established a number of properties of this invariant and conjectured that if the area of is A() then and that equality characterizes triangles. In this note Santaló's conjecture is shown to be true.

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Type
Research Article
Information
Mathematika , Volume 8 , Issue 1 , June 1961 , pp. 63 - 65
Copyright
Copyright © University College London 1961

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References

1.Santaló, L. A., “A new affine invariant of plane and solid convex bodies”, Math. Notae, 16 (1937–38) (Spanish); reviewed in Math. Rev. No. 6567.Google Scholar
2.Ohmann, P., “Extremal probleme für konvexe Bereiche der euklidiscken Ebene”, Math. Zeit., 55 (1951–52), 346–352.CrossRefGoogle Scholar
A reference is given to the dissertation of K. Günther (Marburg 1948). See also Sholander, M., Trans. Amer. Math. Soc., 73 (1952), 139–173.CrossRefGoogle Scholar