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Stability Theorems for Convex Domains of Constant Width

Published online by Cambridge University Press:  20 November 2018

H. Groemer
H. Groemer*
Affiliation:
Department of Mathematics, The University of ArizonaTucson, AZ 85721 USA
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Abstract

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It is known that among all plane convex domains of given constant width Reuleaux triangles have minimal and circular discs have maximal area. Some estimates are given concerning the following associated stability problem: If K is a convex domain of constant width w and if the area of K differs at most ∊ from the area of a Reuleaux triangle or a circular disc of width w, how close (in terms of the Hausdorff distance) is K to a Reuleaux triangle or a circular disc? Another result concerns the deviation of a convex domain M of diameter d from a convex domain of constant width if the perimeter of M is close to πd.

Keywords

52A10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 3 , 01 September 1988 , pp. 328 - 337
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Chakerian, G. D. and Groemer, H., Convex bodies of constant width, in ‘'Convexity and Its Applications” (Gruber, P.M. and Wills, J. M. eds.) Birkhäuser, Basel, Boston, Stuttgart, (1983), pp. 49–96.Google Scholar
2. Bonnesen, T. and Fenchel, W., Théorie der konvexen Körper, Ergebn. d. Math. u. ihrer Grenzgeb. Bd. 3, J. Springer-Verlag Berlin, 1934.Google Scholar
3. Eggleston, H. G., A proof of Blaschke's theorem on the Reuleaux triangle, Quart. J. Math. Oxford, Ser. (2) 3 (1952), pp. 296–297.Google Scholar
4. Eggleston, H. G., Convexity, Cambr. Tracts in Math, and Math. Phys. No. 47. Cambridge Univ. Press, New York, 1959.Google Scholar
5. Osserman, R., A strong form of the isoperimetric inequality in Rn , to appear.Google Scholar
6. Vitale, R. A., Lp metrics for compact convex sets, J. Approx. Theory, 45 (1985), pp. 280–287.Google Scholar