Volume of mixed bodies
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 Trans. Amer. Math. Soc. 294 (1986), 487500 Request permission
Abstract:
By using inequalities obtained for the volume of mixed bodies and the Petty Projection Inequality, (sharp) isoperimetric inequalities are derived for the projection measures (Quermassintegrale) of a convex body. These projection measure inequalities, which involve mixed projection bodies (zonoids), are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body. The inequality obtained for the volume of mixed bodies is also used to derive a form of the BrunnMinkowski inequality involving mixed bodies. As an application, inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies.References

A. D. Aleksandrov, Extension of certain concepts in the theory of convex bodies, Mat. Sb. (N.S.) 2 (1937), 947972. (Russian)
—, New inequalities between mixed volumes and their applications, Mat. Sb. (N.S.) 2 (1937), 12051238. (Russian)
—, Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sb. (N.S.) 3 (1938), 2746. (Russian)
—, Mixed discriminants and mixed volumes, Mat. Sb. (N.S.) 3 (1938), 227251. (Russian)
—, On the surface area function of a convex body, Mat. Sb. (N.S.) 6 (1939), 167174. (Russian)
 Ethan D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323–345. MR 256265, DOI 10.1090/S0002994719690256265X
 T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, SpringerVerlag, BerlinNew York, 1974 (German). Berichtigter Reprint. MR 0344997
 Ju. D. Burago and V. A. Zalgaller, Geometricheskie neravenstva, “Nauka” Leningrad. Otdel., Leningrad, 1980 (Russian). MR 602952
 Herbert Busemann, Convex surfaces, Interscience Tracts in Pure and Applied Mathematics, no. 6, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. MR 0105155
 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/S00029947196702126781
 G. D. Chakerian, The mean volume of boxes and cylinders circumscribed about a convex body, Israel J. Math. 12 (1972), 249–256. MR 317177, DOI 10.1007/BF02790751
 G. D. Chakerian, Isoperimetric inequalities for the mean width of a convex body, Geometriae Dedicata 1 (1973), no. 3, 356–362. MR 322683, DOI 10.1007/BF00147770
 G. D. Chakerian, Geometric inequalities for plane convex bodies, Canad. Math. Bull. 22 (1979), no. 1, 9–16. MR 532264, DOI 10.4153/CMB19790023
 G. D. Chakerian and J. R. SangwineYager, A generalization of Minkowski’s inequality for plane convex sets, Geom. Dedicata 8 (1979), no. 4, 437–444. MR 553681, DOI 10.1007/BF00183259 W. Fenchel and B. Jessen, Mengenfunktionen und konvexe Körper, Danske Vid. Selskab. Mat.Fys. Medd. 16 (1938), 131.
 William J. Firey and Branko Grünbaum, Addition and decomposition of convex polytopes, Israel J. Math. 2 (1964), 91–100. MR 175032, DOI 10.1007/BF02759949
 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 B. Grübaum, Convex polytopes, Interscience, New York, 1967. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934.
 Kurt Leichtweiss, Konvexe Mengen, Hochschultext [University Textbooks], SpringerVerlag, BerlinNew York, 1980 (German). MR 586235
 Erwin Lutwak, Widthintegrals of convex bodies, Proc. Amer. Math. Soc. 53 (1975), no. 2, 435–439. MR 383254, DOI 10.1090/S00029939197503832545
 Erwin Lutwak, Mixed widthintegrals of convex bodies, Israel J. Math. 28 (1977), no. 3, 249–253. MR 464070, DOI 10.1007/BF02759811
 Erwin Lutwak, A general isepiphanic inequality, Proc. Amer. Math. Soc. 90 (1984), no. 3, 415–421. MR 728360, DOI 10.1090/S00029939198407283603
 Erwin Lutwak, Mixed projection inequalities, Trans. Amer. Math. Soc. 287 (1985), no. 1, 91–105. MR 766208, DOI 10.1090/S00029947198507662087
 C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535–1547. MR 133733
 C. M. Petty, Isoperimetric problems, Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971) Dept. Math., Univ. Oklahoma, Norman, Okla., 1971, pp. 26–41. MR 0362057
 Rolf Schneider, Über Tangentialkörper der Kugel, Manuscripta Math. 23 (1977/78), no. 3, 269–278 (German, with English summary). MR 486345, DOI 10.1007/BF01171753 —, Boundary structure and curvature of convex bodies, Contributions to Geometry (J. Tölke and J. M. Wills, eds.), Birkhäser Verlag, Basel, 1979, pp. 159.
 Rolf Schneider, On the AleksandrovFenchel inequality, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 132–141. MR 809200, DOI 10.1111/j.17496632.1985.tb14547.x
 Rolf Schneider and Wolfgang Weil, Zonoids and related topics, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 296–317. MR 731116
Additional Information
 © Copyright 1986 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 294 (1986), 487500
 MSC: Primary 52A40; Secondary 52A22
 DOI: https://doi.org/10.1090/S00029947198608257173
 MathSciNet review: 825717