Minimal fixing systems for convex bodies.

*(English)*Zbl 0867.52002Let \(M\) 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 \(\rho(M)=2n- 1\).

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.) |

##### References:

[1] | Fejes Tóth L., Acta Math. Acad. Sci. Hungar. 13 pp 379– (1962) · Zbl 0113.16203 |

[2] | Boltyanski V., Dokl. AN SSSR. 226 (2) pp 249– (1976) |

[3] | Grunbaum B., Acta Math. Acad. Sci. Hungar. 15 pp 161– (1964) · Zbl 0132.17303 |

[4] | Boltyanski V., Discrete Comput. Geom. 8 pp 27– (1992) · Zbl 0756.52009 |

[5] | Nagy B., Math. 15 (3) pp 169– (1954) |

[6] | Kincses J., Geom. Dedicata 22 pp 283– (1987) |

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