Overview
This paper presents a generalization of Helly's theorem, introducing what are termed 'Overlap-Helly theorems.' These theorems are identified as being closely connected to Bárány-Gromov overlap theorems, also known as selection lemmas.
Research Context
The work directly addresses foundational concepts within geometry and topology, specifically building upon Helly's theorem and its generalizations. It draws a connection to Bárány-Gromov overlap theorems, which are referred to as selection lemmas in the field.
Approach
The research introduces a conceptual generalization of an existing mathematical theorem. It then investigates the implications and connections of this new framework to other established theorems and concepts.
Findings
- A generalization of Helly's theorem, referred to as 'Overlap-Helly theorems,' is introduced.
- These Overlap-Helly theorems are closely connected to Bárány-Gromov overlap theorems (selection lemmas).
- The main result indicates that this new framework implies the topological colorful Helly theorem of Kalai and Meschulam.
- The main result also implies Karasev's topological centerpoint theorem.
- The research further investigates the topological fractional Helly theorem from this overlap perspective.
- An overlap theorem is shown for dense complexes, which is characterized as a continuous second selection lemma for tame maps.
Why This Matters
The introduction of Overlap-Helly theorems provides a unified perspective on several significant results in geometric combinatorics and topology, directly implying established theorems such as the topological colorful Helly theorem and Karasev's topological centerpoint theorem. This framework also offers a new lens for investigating the topological fractional Helly theorem and establishes a continuous second selection lemma for tame maps through an overlap theorem for dense complexes.