Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T01:31:30.490Z Has data issue: false hasContentIssue false
  • English
  • Français

Homothetic ellipsoids1

Published online by Cambridge University Press:  24 October 2008

P. R. Goodey
P. R. Goodey
Affiliation:
Royal Holloway College
Get access Rights & Permissions [Opens in a new window]

Abstract

We let K1 and K2 be two convex bodies in Ed (d ≥ 3) and consider translates of K2 for which ∂K1 ∩ ∂ ≠ ø and K1 ≠ . We show that if, for all such translates,∂K1 ∩ is contained in a hyperplane then K1 and K2 are homothetic ellipsoids.

If K1 and K2 are two planar non-coincident convex sets whose interiors intersect we define α(K1, K2) to be the number of connected components of ∂K1 ∩ ∂K2. The starting point for our investigations is the result of Fujiwara(S) and Bol(2) to the effect that K is a circular disc if and only if α(K, K′) = 2 for all congruent copies K′ of K. An easy extension of this result is the observation that K1 and K2 are congruent circular discs if and only if for all congruent copies of K2. When working with such results it is natural to look for analogues in the situation where only translates are allowed rather than all possible congruences. The appropriate analogue was obtained in (7) and we state it here for completeness.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 1 , January 1983 , pp. 25 - 34
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Aleksandrov, A. D.Zur Theorie der gemischten Volumina von konvexen Körpern, Mat. Sbornik N.S. 2 (1937), 947–972 and 1205–1238; Mat. Sbornik N.S. 3 (1938), 27–46 and 227–251.Google Scholar
(2)Bol, G.Zur kinematischen Ordnung ebener Jordan-Kurven Abh. Math. Sem. Univ. Hamburg 11 (1936), 394–408.CrossRefGoogle Scholar
(3)Bonnesen, T. and Fenchel, W.Theorie der konvexen Körper (New York, 1948).Google Scholar
(4)Burton, G. R.Some characterizations of the ellipsoid. Israel J. Math. 28 (1977), 339–349.CrossRefGoogle Scholar
(5)Fujiwara, M.Ein Satz über konvexe geschlossene Kurven. Sci. Repts. Tóhoku Univ. 9 (1920), 289–294.Google Scholar
(6)Goodey, P. R. Connectivity and freely rolling convex bodies. (Submitted.)Google Scholar
(7)Goodey, P. R. and Woodcock, M. M. Intersections of convex bodies with their translates, The Geometric Vein, ed. Davis, C., Grünbaum, B. and Sherk, F. A. (Springer-Verlag, 1982).Google Scholar
(8)Yanagihara, K.A theorem on surface. Tóhoku Math. J. 8 (1915), 42–44.Google Scholar