# A Helly theorem for an enlarged class of orthogonally starshaped sets in \(\mathbb {R}^d\)

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## Abstract

Consider a finite family \(\mathcal {C}\) of distinct boxes in \(\mathbb {R}^d\), with G the intersection graph of \(\mathcal {C}\) and with \(S \equiv \cup \{C : C \, \hbox {in} \, \mathcal {C}\}\). Assume that *G* is connected and, for each block *B* of *G*, assume that the corresponding members of \(\mathcal {C}\) have a staircase convex union *D*(*B*). When this occurs, we say that the orthogonal polytope *S* *has suitable properties*. Now let \(\mathbf{S}\) be a finite family of orthogonal polytopes such that, for every nonempty subfamily \(\mathbf{S}^{'}\) of \(\mathbf{S}\), the corresponding intersection \(S^{'} \equiv \cap \{S : S \, \hbox {in} \, \mathbf{S}^{'} \}\) (if nonempty) has suitable properties and preserves transitions between certain *D*(*B*) sets. If every \(d +1\) (not necessarily distinct) members of \(\mathbf{S}\) meet in a (nonempty) staircase starshaped set, then \(S_0 \equiv \cap \{S : S \, \hbox {in} \, \mathbf{S} \}\) is nonempty and staircase starshaped as well.

### Keywords

Orthogonal polytope Staircase starshaped set Helly theorem### Mathematics Subject Classification

Primary 52A30 52A35### References

- Bobylev, N.A.: The Helly theorem for star-shaped sets. J. Math. Sci.
**105**, 1819–1825 (2001)MathSciNetCrossRefGoogle Scholar - Breen, M.: A Helly theorem for intersections of orthogonally starshaped sets. Arch. Math.
**80**, 664–672 (2003)MathSciNetCrossRefMATHGoogle Scholar - Breen, M.: A Helly-type theorem for intersections of orthogonally starshaped sets in \({\mathbb{R}}^d\). Period. Math. Hungar.
**68**, 45–53 (2014)MathSciNetCrossRefMATHGoogle Scholar - Breen, M.: A Krasnosel’skii-type theorem for an enlarged class of orthogonal polytopes. Adv. Geom.
**17**(4), 525–532 (2017)MathSciNetCrossRefGoogle Scholar - Breen, M.: An enlarged class of orthogonal polytopes having staircase convex kernels. J. Combin. Math. Combin. Comput.
**101**, 3–12 (2017)MathSciNetMATHGoogle Scholar - Breen, M.: An improved Krasnosel’skii-type theorem for orthogonal polygons which are starshaped via staircase paths. J. Geom.
**51**, 31–35 (1994)MathSciNetCrossRefMATHGoogle Scholar - Chepoi, V.: On staircase starshapedness in rectilinear spaces. Geom. Dedicat.
**63**, 321–329 (1996)MathSciNetCrossRefMATHGoogle Scholar - Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. Convex. In: Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, RI, pp. 101–180 (1962)Google Scholar
- Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, pp. 389–448. North Holland, New York (1993)CrossRefGoogle Scholar
- Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)CrossRefMATHGoogle Scholar
- Krasnosel’skii, M.A.: Sur un critère pour qu’un domaine soit étoilé. Math. Sb. (61)
**19**, 309–310 (1946)MATHGoogle Scholar - Lay, S.R.: Convex Sets and Their Applications. Wiley, New York (1982)MATHGoogle Scholar
- Valentine, F.A.: Convex Sets. McGraw-Hill, New York (1964)MATHGoogle Scholar