References
O. Aberth,An isoperimetric inequality. Proc. London Math. Soc. (3)13 (1963), 322–336.
P. Alexandroff undA. Hopf,Topologie I. Springer, Berlin, 1935.
V. Borovikov,On the intersection of a sequence of simplices (Russian). Uspehi Mat. Nauk7, No. 6 (52), (1952) 179–180.
J. J. A. M. Brands andG. Laman,Plane section of a tetrahedron. Amer. Math. Monthly70 (1963), 338–339.
G. Choquet,Unicité des representations intégrales dans les cônes convexes. C. R. Acad. Sci. Paris243 (1956), 699–702.
G. Choquet andP.-A. Meyer,Existence et unicité des représentations intégrales dans les convexes compacts quelconque. Ann. Inst. Fourier, Grenoble13 (1963) 135–154.
H. T. Croft,Some geometrical, thoughts. Math. Gaz., to appear.
H. G. Eggleston,Plane section of a tetrahedron. Amer. Math. Monthly70 (1963) 1108.
L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum. Springer, Berlin, 1953.
David Gale,On the number of faces of a convex polyhedron. Canad. J. Math.16 (1964), 12–17.
H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin, 1957.
W. Hurewicz andH. Wallman,Dimension Theory. Princeton Univ. Press, Princeton (N. J.), 1948.
D. G. Kendall,Simplexes and vector lattices. J. London Math. Soc.37 (1962), 365–371.
V. Klee,On the number of vertices of a convex polytope. Canad. J. Math.16 (1964), 701–720.
V. Klee,Diameters of polyhedral graphs. Canad. J. Math.16 (1964), 602–614.
V. Klee,A property of d-polyhedral graphs. J. Math. Mech., to appear.
V. Klee,Heights of convex polytopes. J. Math. Anal. Appl., to appear.
G. Nöbeling,Über die Länge der euklidischen Kontinuen. Jber. Deutsch. Math. Verein.52 (1942), 189–197.
C. A. Rogers andG. C. Shephard,The difference body of a convex body. Arch. Math.8 (1957), 220–233.
H. Weyl,Elementare Theorie der konvexen Polyeder. Comment. Math. Helv.7 (1935), 290–306.
Additional information
This paper was written at the University of Washington (Seattle, Washington, U.S.A.) while ProfessorsEggleston andGrünbaum were visiting there, on leave respectively from Bedford College, London and The Hebrew University, Jerusalem.Eggleston’s work was supported by a fellowship from the National Science Foundation (U. S. A.),Grünbaum’s andKlee’s by an N. S. F. grant (NSF-GP-378).
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Eggleston, H.G., Grünbaum, B. & Klee, V. Some semicontinuity theorems for convex polytopes and cell-complexes. Commentarii Mathematici Helvetici 39, 165–188 (1964). https://doi.org/10.1007/BF02566949
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DOI: https://doi.org/10.1007/BF02566949