Minimal Fixing Systems for Convex Bodies
نویسندگان
چکیده
L. Fejes Tóth [1] introduced the notion of fixing system for a compact, convex body M ⊂ Rn. Such a system F ⊂ bd M stabilizes M with respect to translations. In particular, every minimal fixing system F is primitive, i.e., no proper subset of F is a fixing system. In [2] lower and upper bounds for cardinalities of mimimal fixing systems are indicated. Here we give an improved lower bound and show by examples, now both the bounds are exact. Finally, we formulate a Fejes Tóth Problem. 1. Main definitions and former results. Let M be a compact, convex body and F ⊂ bd M. A direction l defined by a nonzero vector e ∈ Rn is said to be an outer moving direction with respect to F if for any λ > 0, the relation (−λe + int M) ∩ F = ∅ holds. A set F ⊂bdM is a fixing system for M if there is no outer moving direction with respect to F. Visually, F is a fixing system for M if, assuming ”fixing nails” at all points of F, it is impossible to translate M in any direction. A fixing system F ⊂ bd M is primitive if no proper subset of F is a fixing system for M. We denote by %(M) the smallest of the integers s 1991 Mathematics Subject Classification. 52A20, 52A37, 52B05.
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