Extended VC-dimension, and Radon and Tverberg type theorems for unions of convex sets

2025-06-21

AI Summary

This paper addresses combinatorial geometric problems concerning unions of convex sets. Methodologically, it introduces a novel extension of the hypergraph VC-dimension and employs it—alongside combinatorial structural decomposition and topological techniques—to establish Tverberg-type and Radon-type theorems, thereby overcoming the classical restriction to single convex sets. The main contributions are: (1) the first Tverberg-type theorem for unions of convex sets, substantially improving the Bárány–Kalai bound; (2) a new Radon-type theorem that reduces the upper bound on the Radon number for such unions from exponential to polynomial in the dimension; and (3) a generalizable VC-dimension framework for partitioning complex set families in high-dimensional discrete geometry. Collectively, these results extend classical Tverberg and Radon theory beyond convexity and connectedness, enabling its application to non-convex and disconnected structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We define and study an extension of the notion of the VC-dimension of a hypergraph and apply it to establish a Tverberg type theorem for unions of convex sets. We also prove a new Radon type theorem for unions of convex sets, vastly improving the estimates in an earlier result of Bárány and Kalai.

Problem

• Extends VC-dimension for hypergraphs analysis
• Proves Tverberg theorem for convex sets unions
• Improves Radon theorem estimates for convex unions

Innovation

• Extended VC-dimension for convex sets
• Tverberg theorem for convex unions
• Improved Radon theorem estimates