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Minimal fixing systems for convex bodies. (English) Zbl 0867.52002
Let $$M$$ be a convex body in $$R^n$$ and let $$F$$ be a subset of the boundary of $$M$$. A unit sector $$v\in R^n$$ defines an outer moving direction with respect to $$F$$ if, for any $$\lambda> 0$$, the translate $$M- \lambda v$$ of $$M$$ does not contain any point of $$F$$ in its interior. The set $$F$$ is called a fixing system for $$M$$ if there is no outer moving direction with respect to $$F$$. Equivalently, $$F$$ is a fixing system for $$M$$ if and only if each direction illuminates at least one point of $$F$$. The cardinality $$\rho(M)$$ of the fixing systems for $$M$$ consisting of the minimum number of points has been studied by V. Boltyanski and H. Martini in Results Math. 28, No. 3-4, 224-249 (1995; Zbl 0853.52004), where the following lower and upper bounds have been obtained: $n+1\leq \rho(M)\leq 2n+ 1 -\text{md }M;$ here $$\text{md }M$$ denotes the maximum integer $$m$$ such that there are $$m+ 1$$ regular boundary points of $$M$$ with outer unit normals $$v_0,\dots,v_m$$ which are vertices of an $$m$$-dimensional simplex containing the origin in its interior.
In the present paper the authors improve the lower bound in the previous inequality by showing that $n+\frac{n}{\text{md }M}\leq \rho(M).$ They also prove that for any integer $$\rho$$ satisfying the inequality: $n+\frac{n}{m}\leq \rho(M)\leq 2n+ 1- m,$ where $$1\leq m\leq n$$, there exists a convex body $$M$$ with $$\text{md }M= m$$ and $$\rho(M)=\rho$$. Finally, they describe geometrically all the convex bodies $$M\subset R^n$$ with $$\rho(M)= 2n$$ and $$\rho(M)=2n- 1$$.
Reviewer: C.Peri (Milano)

##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 52A37 Other problems of combinatorial convexity 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
##### Keywords:
fixing system; illumination; combinatorial geometry
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##### References:
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