Transversal Helly numbers, pinning theorems and projection of simplicial complexes

Xavier Goaoc 1
1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : The efficient resolution of various problems in computational geometry, for instance visibility computation or shape approximation, raises new questions in line geometry, a classical area going back to the mid-19th century. This thesis fits into this theme, and studies Helly numbers of certain sets of lines, an index related to certain basis theorems arising in computational geometry and combinatorial optimization. Formally, the Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. For $d\ge 2$ let $h_d$ denote the least integer such that for any family $\{B_1, \ldots, B_n\}$ of pairwise disjoint balls of equal radius in $R^d$, the Helly number of $\{T(B_1), \ldots, T(B_n)\}$ is at most $h_d$, where $T(B_i)$ denotes the set of lines intersecting $B_i$. In 1957, Ludwig Danzer showed that $h_2$ equals $5$ and conjectured that $h_d$ is finite for all $d \ge 2$ and increases with $d$. We establish that $h_d$ is at least $2d-1$ and at most $4d-1$ for any $d \ge 2$, proving the first conjecture and providing evidence in support of the second one. To study Danzer's conjectures, we introduce the pinning number, a local analogue of the Helly number that is related to grasping questions studied in robotics. We further show that pinning numbers can be bounded for sufficiently generic families of polyhedra or ovaloids in $R^3$, two situations where Helly numbers can be arbitrarily large. A theorem of Tverberg asserts that when $\{B_1, \ldots, B_n\}$ are disjoint translates of a convex figure in the plane, the Helly number of $\{T(B_1), \ldots, T(B_n)\}$ is at most $5$. Although quite different, both our and Tverberg's proofs use, in some way, that the intersection of at least two $T(B_i)$'s has a bounded number of connected components, each contractible. Using considerations on homology of projection of simplicial complexes and posets, we unify the two proofs and show that such topological condition suffice to ensure explicit bounds on Helly numbers.
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Xavier Goaoc. Transversal Helly numbers, pinning theorems and projection of simplicial complexes. Computational Geometry [cs.CG]. Université Henri Poincaré - Nancy I, 2011. ⟨tel-00650204v2⟩

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