Abstract
In this paper we consider a new class of convex bodies which was introduced in [11]. This is the class of belt bodies, and it is a natural generalization of the class of zonoids (see the surveys [18, 28, 24]). While the class of zonoids is not dense in the family of all centrally symmetric, convex bodies, the class of belt bodies is dense in the set of all convex bodies. Nevertheless, we shall extend solutions of combinatorial problems for zonoids (cf. [2, 12]) to the class of belt bodies. Therefore, we first introduce the set of belt bodies by using zonoids as starting point. (To make the paper self-contained, a few parts of the approach from [11] are given repeatedly.) Second, complete solutions of three well-known (and generally unsolved) problems from the combinatorial geometry of convex bodies are given for the class of belt bodies. The first of these, connected with the names of I. Gohberg and H. Hadwiger, is the problem of covering a convex body with smaller homothetic copies, or the equivalent illumination problem. The second is the Szökefalvi-Nagy problem, which asks for the determination of the convex bodies whose families of translates have a given Helly dimension. The third problem concerns special fixing systems, a notion which is due to L. Fejes Tóth. These solutions consist of improved and more general approaches to recently solved problems (as in the case of the Helly-dimensional classification of belt bodies) or new results (as those concerning minimal fixing systems, providing also an answer to a problem of B. Grünbaum which is not only restricted to belt bodies).
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References
E. Baladze: Complete solution of the Szökefalvi-Nagy problem for zonotopes. Soviet Math. Dokl. 34 (1987), no. 3, 458–461.
E. Baladze: A solution of the Szökefalvi-Nagy problem for zonoids (Russian). Doklady AN SSSR, 310 (1990), no. 1, 11–14.
E. Baladze: Solution of the Szökefalvi-Nagy problem for a class of convex polytopes. Geom. Dedicata 49 (1994), 25–38.
E. Baladze and V. Boltyanski: Belt bodies and the Helly dimension (Russian). Mat. Sbornik (to appear).
E.D. Bolker: A class of convex bodies. Trans. Amer. Math. Soc. 145 (1969), 323–345.
E.D. Bolker: Centrally symmetric polytopes. In: Proc. 12th Biannual Intern. Semin. Canad. Math. Congr. on Time Series and Stochastic Processes, Convexity and Combinatorics (Vancouver 1969), Ed. R. Pyke, Canad. Math. Congress, Montreal 1970, 255–263
B. Bollobás: Fixing systems for convex bodies. Studia Sci. Math. Hungar. 2 (1967), 351–354.
V. Boltyanski: A problem of illuminating the boundary of a convex body (Russian). Izvestija Mold. Fil. AN SSSR 10(76) (1960), 77–84.
V. Boltyanski: The theorem of Helly for H-convex sets. Dokl. AN SSSR 226 (1976), no. 2, 249–252.
V. Boltyanski: A new step in the solution of the Szökefalvi-Nagy problem. Discrete Cornput. Geom. 8 (1992), 27–49.
V. Boltyanski: A solution of the illumination problem for belt bodies. Mat. Zametki (to appear).
V. Boltyanski and P. Soltan: A solution of Hadwiger’s covering problem for zonoids. Combinatorica 12(4) (1992), 381–388.
L. Danzer: Math. Reviews 2942 (1963), 569–570.
L. Fejes Tóth: On primitive polyhedra. Acta Math. Acad. Sci. Hungar. 13 (1952), 379–382.
S. Fudali: Six-point primitive system in a plane. Demonstratio Math. 19 (1986), no. 2, 341–348.
S. Fudali: Fixing system and homothetic covering. Acta Math. Hungar. 50 (1987), no. 3–4, 203–225.
I. Gohberg and A. Markus: A certain problem about the covering of convex sets with homothetic ones (in Russian). Izvestija Mold. Fil. AN SSSR 10(76) (1960), 87–90.
P.R. Goodey, W. Weil: Zonoids and generalizations. In: Handbook of Convex Geometry, Eds. P.M. Gruber and J.M. Wills, North-Holland, 1993, 1297–1326.
B. Grünbaum: Fixing systems and inner illumination. Acta Math. Acad. Sci. Hungar. 15 (1964), 161–163.
H. Hadwiger: Ungelöste Probleme, Nr. 20. Eiern. Math. 12 (1957), 121.
H. Hadwiger: Ungelöste Probleme, Nr. 38. Eiern. Math. 15 (1960), 130–131.
P. Mani: On polytopes fixed by their vertices. Acta Math. Acad. Sci. Hungar. 22 (1971), 269–273.
H. Martini: Some results and problems around zonotopes. Coll. Math. Soc. J. Bolyai 48 (1985), 383–418, North-Holland, Amsterdam et al., 1987.
H. Martini: Cross-sectional measures. Coll. Math. Soc. J. Bolyai 63 (1992), 269–310, North-Holland, Amsterdam et al., 1994.
S. Saks: Theory of the Integral. Hafner, New York, 1937.
R. Schneider: On the Aleksandrov-Fenchel inequality involving zonoids. Geom. Dedicata 27 (1988), 113–126.
R. Schneider: Convex Bodies: the Brunn-Minkowski Theory. Cambridge Univ. Press, Cambridge 1993.
R. Schneider, W. Weil: Zonoids and related topics. In: Convexity and Its Applications, Eds. P.M. Gruber and J.M. Wills, Birkhäuser, Basel 1983, 296–317.
B. Szökefalvi-Nagy: Ein Satz über Parallelverschiebungen konvexer Körper. Acta Sci. Math. 15 (1954), no. 3–4), 169–177.
J.E. Taylor: Zonohedra and generalized zonohedra. Amer. Math. Monthly 99 (1992), 108–111.
B. Tomor: The fixing problem for convex figures (in Hungarian). Mat. Lapok 14 (1963), 120–123.
H.S. Witsenhausen: A support characterization of zonotopes. Mathematika 25 (1978), 13–16.
V. Zalgaller and Ju. Reshetnyak: On rectifiable curves, additive vector function and displacement of segments. Vestnik Leningrad. Univ., Mat. Fiz. Kim, 9 (1954), no. 2, 45–67 (in Russian).
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Research supported by “Deutsche Forschungsgemeinschaft”
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Boltyanski, V., Martini, H. Combinatorial geometry of belt bodies. Results. Math. 28, 224–249 (1995). https://doi.org/10.1007/BF03322255
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DOI: https://doi.org/10.1007/BF03322255