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) arid each body isometric to L(C) is illuminated by D. Denote by En the minimum i...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) arid 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.展开更多
In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-coverin...In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger＇s covering conjecture and Borsuk＇s partition conjecture. They are flmdamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.展开更多
文摘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) arid 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.
基金Supported by 973 Programs(Grant Nos.2013CB834201 and 2011CB302401)the National Science Foundation of China(Grant No.11071003)the Chang Jiang Scholars Program of China
文摘In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger＇s covering conjecture and Borsuk＇s partition conjecture. They are flmdamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.
