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The Steinitz-Gross Theorem on Sums of Vectors

Published online by Cambridge University Press:  20 November 2018

F. A. Behrend
F. A. Behrend*
Affiliation:
University of Melbourne
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α1α2, …, αp are n-dimensional vectors,

,;

they are arranged to form a closed polygon

.

Denote by R(α1, α2, …, αp) the radius of the smallest circumscribed hypersphere with centre at 0 ; by R(α1, α2, …, αp) the minimum of

(α1, α2, …, αp)

for all possible reorderings

of α2, …, αp−1 and by cn the least possible constant such that

for all possible choices of p and α1α2, … , αp.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 108 - 124
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Steinitz, E., Bedingt Convergente Reihen und konvexe Systeme, J. reine angew. Math. 143 (1913), 128–175; 144 (1914), 1–40.Google Scholar
2. Gross, W., Bedingt Convergente Reihen, Monatsh. Math. Phys. 28 (1917), 221–237.Google Scholar
3. Bergström, V., Ein neuer Beweis eines Satzes von E. Steinitz, Abh. Math. Seminar Hamburg, 8 (1930), 148–152.Google Scholar
4. Bergström, V., Zwei Sätze über ebene Vektor poly gone, Abh. Math. Seminar Hamburg, 8 (1930), 206–214.Google Scholar
5. Damsteeg, I. and Halperin, I., The Steinitz–Gross theorem on sums of vectors, Trans. Roy. Soc. Can., sec. III, 44 (1950), 31–35.Google Scholar