ILLUMINATION AND EXPOSITION OF A CONVEX n-DIMENSIONAL BODY DEPENDING ON ITS SHARPNESS

摘要/Abstract

摘要：

A convex n-body C is said to be exposable to a set D of a few directions if there is a linear transformation L : En ! En such that L(C) and each body isometric to L(C) is illuminated by D. Denote by En the minimum integer such that each C is exposable to a (fixed) set D of cardinality En. An upper bound for En is established here.

关键词: Covering, illumination, Hadwiger conjecture

Abstract:

Key words: Covering, illumination, Hadwiger conjecture

中图分类号:

52C17

Boris V. Dekster. ILLUMINATION AND EXPOSITION OF A CONVEX n-DIMENSIONAL BODY DEPENDING ON ITS SHARPNESS[J]. 数学物理学报(英文版), 2002, 22(2): 189-198.

Boris V. Dekster. ILLUMINATION AND EXPOSITION OF A CONVEX n-DIMENSIONAL BODY DEPENDING ON ITS SHARPNESS[J]. Acta mathematica scientia,Series B, 2002, 22(2): 189-198.

参考文献

[1]Boltjansky V G, Gohberg I Ts. Results and problems in combinatorial geometry. Cambridge: Cambridge University Press, 1985

[2]Boltyanski V G. The problem of the illumination of the boundary of a convex body. (Russian) Izv Mold Fil Acad Nauk SSSR, 1960,10(76): 77-84

[3]Boltyanski V, Martini H, Soltan P S. Excursions into Combinatorial Geometry. Springer, 1997

[4]Gruber P M.Aspects of approximation of convex bodies. In: Gruber P M, Wills J M, eds. Handbook of convex geometry. North-Holland: Elsevier Science Publishers B V, 1993

[5]Hadwiger H. ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers. Comment Math Helv,1945/46,18: 73-75

[6]Hadwiger H. Ungel¨oste Problem. Nr 20 Elem Math, 1957,12: 121

[7]Hadwiger H. Ungel¨oste Problem. Nr 38 Elem Math, 1960,15: 130-131

[8]John, F. Extremum problems with inequalities as subsidiary conditions. In: Courant Anniversary Volume.New York: Interscience, 1948. 187-204

[9]Lassak M. Covering the boundary of a convex set by tiles. Proceedings of AMS, 1988,104: 269-272

[10]Lassak M. Covering a three-dimensional convex body by smaller homothetic copies. Beitr Algebra Geom,1998, 39(2): 259-262

[11]Lindenstrauss J, Milman V D. The local theory of normed spaces and its applications to convexity. In:Gruber P M,Wills J M, eds. Handbook of convex geometry. North Holland: Elsevier Science Publishers B V, 1993

[12]Martini H. Some results and problems around zonotopes. Coloquia Mathematica Societatis Janos Bolyai 48. Intuitive geometry, Siofok, 1985. 383-418

[13]Rankin R A. The closest packing of spherical caps in n dimensions. Proceedings of Glasgow Math Assoc,1955,(2):139-144

[14]Rogers C A, Zong C. Covering convex bodies by translates of convex bodies. Mathematika, 1997, 44(1):215-218

[15]Schmitt P. Problems in discrete and combinatorial geometry. In: Gruber P M, Wills J M, eds. Handbook of geometry. North Holland: Elsevier Science Publishers B V, 1993

[16]Weissbach B. Eine Bemerkung zur ¨Uberdeckung beschrankter Mengen durch Mengen kleineren Durchmes-sers. Beitr Algebra Geom, 1981,11: 119-122

[17]Wyner A D. Random packings and coverings of the unit n-sphere. Bell System Tech Journal, 1967, 46:2111-2118