Proof of Gru¨nbaum's conjecture on common transversals for translates
Discrete & Computational Geometry
Helly-type theorems and geometric transversals
Handbook of discrete and computational geometry
Geometric permutations of balls with bounded size disparity
Computational Geometry: Theory and Applications - Special issue on the thirteenth canadian conference on computational geometry - CCCG'01
Intersections of Leray complexes and regularity of monomial ideals
Journal of Combinatorial Theory Series A
Inclusion-Exclusion Formulas from Independent Complexes
Discrete & Computational Geometry
Line Transversals to Disjoint Balls
Discrete & Computational Geometry
Helly-Type Theorems for Line Transversals to Disjoint Unit Balls
Discrete & Computational Geometry
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Geometric permutations of disjoint unit spheres
Computational Geometry: Theory and Applications
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Efficient data structure for representing and simplifying simplicial complexes in high dimensions
Proceedings of the twenty-seventh annual symposium on Computational geometry
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
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The nerve of a family of sets is a simplicial complex that records the intersection pattern of its subfamilies. Nerves are widely used in computational geometry and topology, because the nerve theorem guarantees that the nerve of a family of geometric objects has the same topology as the union of the objects, if they form a good cover. In this paper, we relax the good cover assumption to the case where each subfamily intersects in a disjoint union of possibly several homology cells, and we prove a generalization of the nerve theorem in this framework, using spectral sequences from algebraic topology. We then deduce a new topological Helly-type theorem that unifies previous results of Amenta, Kalai and Meshulam, and Matousek. This Helly-type theorem is used to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory.