Abstract
We consider several questions from the Brunn-Minkowski theory of convex bodies, concentrating on problems that can be treated successfully if restricted to polytopes. Topics under investigation are a variant of indecomposability with respect to Minkowski addition, addition and decomposition problems involving intermediate area measures, determination by volumes of projections, special representations of mixed volumes, and the equality problem for the Aleksandrov-Fenchel inequality
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References
Betke U. 1992, Mixed volumes of poly topes. Arch. Math. 58 (1992), 388–391
Ewald G. 1990, A short proof of Alexandrov-Fenchel’s inequality. Note di Matematica 10, Suppl. n. 1 (1990), 243–249
Ewald G. and Tondorf E. 1990+, A contribution to equality in Alexandrov-Fenchel’s inequality. Preprint 1990 (and lecture at Oberwolfach, July 1990)
Fedotov V. P. 1979, A counterexample to a hypothesis of Firey. (Russian) Mat. Zametki 26 (1979), 269–275. English translation: Math. Notes 26 (1979), 626-629
Fedotov V. P. 1982, Polar representation of a convex compactum. (Russian) Ukrain. Geom. Sb. 25 (1982), 137–138
Firey W. J. 1970, Convex bodies of constant outer p-measure. Mathematika 17 (1970), 21–27
Firey W. J. and Grünbaum B. 1964, Addition and decomposition of convex polytopes. Israel J. Math. 2 (1964), 91–100
Gardner R. J and Volčič A. 1993+, Determination of convex bodies by their brightness functions, (to appear)
Goodey P. R. and Schneider R. 1980, On the intermediate area functions of convex bodies. Math. Z. 173 (1980), 185–194
Goodey P. R., Schneider R. and Weil W. 1993+, Projection functions on higher rank Grassmannians. In: Geometric Aspects of Functional Analysis (eds. J. Lindenstrauss & V. D. Milman), Lecture Notes in Math., Springer, Heidelberg etc. (to appear)
McMullen P. 1993, On simple polytopes. Invent. Math. 113 (1993), 419–444
McMullen P. and Schneider R. 1983, Valuations on convex bodies. In: Convexity and its Applications (eds. P. M. Gruber & J. M. Wills), Birkhäuser, Basel 1983, pp. 170–247
Pedersen P. and Sturmfels B. 1992+, Product formulas for resultants and Chow forms. Math. Z. (to appear)
Schneider R. 1970, On the projections of a convex polytope. Pacific J. Math. 32 (1970), 799–803
Schneider R. 1993, Convex Bodies: the Brunn-Minkowski Theory. (Encyclopedia of Math, and its Appl., vol. 44) Cambridge University Press, Cambridge 1993
Schneider R. 1991+, Equality in the Aleksandrov-Fenchel inequality — present state and new results. In: Colloquia Math. Soc. János Bolyai 63 (Intuitive Geometry, Szeged 1991) (to appear)
Shephard G. C. 1963, Decomposable convex polyhedra. Mathematika 10 (1963), 89–95
Shephard G. C. 1966, Reducible convex sets. Mathematika 13 (1966), 49–50
Yost D. 1991, Irreducible convex sets. Mathematika 38 (1991), 134–155
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© 1994 Springer Science+Business Media Dordrecht
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Schneider, R. (1994). Polytopes and Brunn-Minkowski Theory. In: Bisztriczky, T., McMullen, P., Schneider, R., Weiss, A.I. (eds) Polytopes: Abstract, Convex and Computational. NATO ASI Series, vol 440. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0924-6_13
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DOI: https://doi.org/10.1007/978-94-011-0924-6_13
Publisher Name: Springer, Dordrecht
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