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Some extremal problems for convex bodies

Published online by Cambridge University Press:  26 February 2010

C. A. Rogers  and
G. C. Shephard
C. A. Rogers
Affiliation:
University of Birmingham, Birmingham, 15.
G. C. Shephard
Affiliation:
University of Birmingham, Birmingham, 15.
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If K is a convex body in n-dimensional space, let SK denote the closed n-dimensional sphere with centre at the origin and with volume equal to that of K. If H and K are two such convex bodies let C(H, K) denote the least convex cover of the union of H and K, and let V*(H, K) denote the maximum, taken over all points x for which the intersection is not empty, of the volume

of the set . The object of this paper is to discuss some of the more interesting consequences of the following general theorem.

Type
Research Article
Information
Mathematika , Volume 5 , Issue 2 , December 1958 , pp. 93 - 102
Copyright
Copyright © University College London 1958

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References

page 93 note † Since the choice of a may not be unique, the symbol R*K may not denote a unique set; no confusion should arise as all the possible sets, which R*K may denote, have the same volume.

page 93 note ‡ Rogers, C. A. and Shephard, G. C., Journal London Math. Soc., 33 (1958), 270–281.CrossRefGoogle Scholar

page 95 note † Loc. cit., 210–211.

page 95 note ‡ Proc. Cambridge Phil. Soc., 47 (1951), 245–247.CrossRefGoogle Scholar

page 95 note § We define the least convex cover to be the least convex set containing the given set; it is not necessarily closed.

page 95 note ║ See, for example, Bonnesen-Fenchel, , Theorie der konvexen Körper (Ergebnisse der Math.), §11.Google Scholar

page 97 note † See, for example, Eggleston, H. G., Convexity (Cambridge 1958), 98.CrossRefGoogle Scholar

page 97 note ‡ Loc. cit., Satz 4.

page 97 note § Loc. cit., Lemma 2.

page 97 note ║ Eggleston, H. G., loc. cit., Ch. 5.Google Scholar

page 98 note † This supposition is convenient for the purely technical reason that it enables us to take the index set I introduced below to be the union of H and K.

page 100 note † Loc. cit.

page 101 note † Macbeath, A. M., Canadian J. of Math., 3 (1951), 54–61.CrossRefGoogle Scholar