A class of convex bodies
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References
-
A. D. Alexandrov, A theorem on convex polyhedra, Trudy Mat. Inst. Steklov. Sect. Math. 4 (1933), 87. (Russian)
—, Zur Theorie der gemischten Volumina von Konvexen Körpern, II, Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen, Mat. Sb. 2 (1937), 1205-1238. (Russian)
W. Blaschke, Vorlesungen über Differential-geometrie. Vol. II, Affine Differentialgeometrie, Springer-Verlag, Berlin, 1923.
- Wilhelm Blaschke, Kreis und Kugel, Walter de Gruyter & Co., Berlin, 1956 (German). 2te Aufl. MR 0077958, DOI 10.1515/9783111506937 W. Blaschke and K. Reidemeister, Über die Entwicklung der Affingeometrie, Jber Deutsch. Math.-Verein. 31 (1922), 63-82.
- Ethan D. Bolker, Functions resembling quotients of measures, Trans. Amer. Math. Soc. 124 (1966), 292–312. MR 197671, DOI 10.1090/S0002-9947-1966-0197671-9
- T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin-New York, 1974 (German). Berichtigter Reprint. MR 0344997
- Joh. Jak Burckhardt, Über konvexe Körper mit Mittelpunkt, Vierteljschr. Naturforsch. Ges. Zürich 85 (1940), no. Beiblatt (Festschrift Rudolf Fueter), 149–154 (German). MR 4986
- G. D. Chakerian, Sets of constant relative width and constant relative brightness, Trans. Amer. Math. Soc. 129 (1967), 26–37. MR 212678, DOI 10.1090/S0002-9947-1967-0212678-1
- Gustave Choquet, Mesures coniques, affines et cylindriques: structure et opérations, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A567–A569 (French). MR 231169
- Gustave Choquet, Mesures coniques et affines invariantes par isométries Zonoformes, zonoèdres et fonctions de type négatif, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A619–A621 (French). MR 231170
- H. S. M. Coxeter, The classification of zonohedra by means of projective diagrams, J. Math. Pures Appl. (9) 41 (1962), 137–156. MR 141004
- H. S. M. Coxeter, Regular polytopes, 2nd ed., The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151873
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- H. G. Eggleston, Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. MR 0124813, DOI 10.1017/CBO9780511566172 W. Fenchel and B. Jessen, Mengenfunctionen und konvexe Körper, Det. Kgl. Danske Vid. Selsk. Math.-Fys. Medd. 3 (1938). W. J. Firey, The brightness of convex bodies, Technical Report No. 19, Department of Mathematics, Oregon State University, Corvallis, Oregon, 1965.
- William J. Firey, Blaschke sums of convex bodies and mixed bodies, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 94–101. MR 0220165 S. Gelbart, ${L^1}$-unit balls and conical measures, Proc. Amer. Math. Soc. (to appear).
- B. Grünbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960), 451–465. MR 114110, DOI 10.1090/S0002-9947-1960-0114110-9
- Branko Grünbaum, Convex polytopes, Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. MR 0226496
- Paul R. Halmos, The range of a vector measure, Bull. Amer. Math. Soc. 54 (1948), 416–421. MR 24963, DOI 10.1090/S0002-9904-1948-09020-6
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2
- C. S. Herz, A class of negative-definite functions, Proc. Amer. Math. Soc. 14 (1963), 670–676. MR 158251, DOI 10.1090/S0002-9939-1963-0158251-7
- Shizuo Kakutani, Concrete representation of abstract $(L)$-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523–537. MR 4095, DOI 10.2307/1968915
- Shizuo Kakutani, Concrete representation of abstract $(M)$-spaces. (A characterization of the space of continuous functions.), Ann. of Math. (2) 42 (1941), 994–1024. MR 5778, DOI 10.2307/1968778
- Robert P. Kaufman and Neil W. Rickert, An inequality concerning measures, Bull. Amer. Math. Soc. 72 (1966), 672–676. MR 193209, DOI 10.1090/S0002-9904-1966-11551-3
- A. Liapounoff, Sur les fonctions-vecteurs complètement additives, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 465–478 (Russian, with French summary). MR 0004080
- Joram Lindenstrauss, On the extension of operators with a finite-dimensional range, Illinois J. Math. 8 (1964), 488–499. MR 169062
- Joram Lindenstrauss, A short proof of Liapounoff’s convexity theorem, J. Math. Mech. 15 (1966), 971–972. MR 0207941
- J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326 N. Lindquist, Representations of central convex bodies, Thesis, Oregon State University, Corvallis, Oregon, 1968.
- E. Mayer, Größte Polygone mit gegebenen Seitenvektoren, Comment. Math. Helv. 10 (1937), no. 1, 288–301 (German). MR 1509581, DOI 10.1007/BF01214297
- Herbert Naumann, Beliebige konvexe Polytope als Schnitte und Projektionen höherdimensionaler Würfel, Simplizes und Masspolytope, Math. Z. 65 (1956), 91–103 (German). MR 78701, DOI 10.1007/BF01473872
- R. E. A. C. Paley, Some theorems on abstract spaces, Bull. Amer. Math. Soc. 42 (1936), no. 4, 235–240. MR 1563277, DOI 10.1090/S0002-9904-1936-06277-4
- C. M. Petty, Projection bodies, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 234–241. MR 0216369
- Neil W. Rickert, The range of a measure, Bull. Amer. Math. Soc. 73 (1967), 560–563. MR 209435, DOI 10.1090/S0002-9904-1967-11747-6
- Neil W. Rickert, Measures whose range is a ball, Pacific J. Math. 23 (1967), 361–371. MR 222244, DOI 10.2140/pjm.1967.23.361
- D. Rutovitz, Some parameters associated with finite-dimensional Banach spaces, J. London Math. Soc. 40 (1965), 241–255. MR 190708, DOI 10.1112/jlms/s1-40.1.241
- Rolf Schneider, Zur einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 71–82 (German). MR 218976, DOI 10.1007/BF01135693
- Rolf Schneider, Über eine Integralgleichung in der Theorie der konvexen Körper, Math. Nachr. 44 (1970), 55–75 (German). MR 275286, DOI 10.1002/mana.19700440105
- Gideon Schwarz, Variations on vector measures, Pacific J. Math. 23 (1967), 373–375. MR 220894, DOI 10.2140/pjm.1967.23.373
- Z. Semadeni, Generalizations of Bohr’s theorem on Fourier series with independent characters, Studia Math. 23 (1963), 159–179. MR 169001, DOI 10.4064/sm-23-2-159-179
- G. C. Shephard, Decomposable convex polyhedra, Mathematika 10 (1963), 89–95. MR 172176, DOI 10.1112/S0025579300003995
- G. C. Shephard, Approximation problems for convex polyhedra, Mathematika 11 (1964), 9–18. MR 172177, DOI 10.1112/S0025579300003430
- G. C. Shephard, Polytopes with centrally symmetric faces, Canadian J. Math. 19 (1967), 1206–1213. MR 221381, DOI 10.4153/CJM-1967-109-3
- Gustave Choquet, Mesures coniques, affines et cylindriques, Symposia Mathematica, Vol. II (INDAM, Rome, 1968) Academic Press, London, 1969, pp. 145–182 (French). MR 0261306
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 145 (1969), 323-345
- MSC: Primary 52.30
- DOI: https://doi.org/10.1090/S0002-9947-1969-0256265-X
- MathSciNet review: 0256265