Minimal fixing systems for convex bodies

V.G. Boltyanski and E. Morales Amaya

Abstract: L. Fejes Tóth [1] introduced the notion of {\it fixing system} for a compact, convex body $\, M\subset R^n.\, $ Such a system $\, F\subset \bd \, M\, $ stabilizes $\, M\, $ with respect to translations. In particular, every {\it minimal} fixing system $\, F\, $ is {\it 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 {\it Fejes Tóth Problem.}

Keywords: Convex body, fixing system, illumination, minimal dependency, functional $md$, indecomposable bodies, combinational geometry

Classification (MSC2000): 52A20, 52A37, 52B05

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