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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)

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.)
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