Quantitative Tverberg, Helly, & Carathéodory theorems

Abstract

This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.

33 pages

References in corpus (3)

• Counterexamples to the topological Tverberg conjecture
• Optimal bounds for the colored Tverberg problem
• A Quantitative Doignon-Bell-Scarf Theorem

Cited by in corpus (6)

• Quantitative $(p,q)$ theorems in combinatorial geometry
• Helly numbers of Algebraic Subsets of ${\mathbb{R}}^d$
• Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of $S$-optimization
• Algorithms for Tverberg's theorem via centerpoint theorems
• Quantitative Tverberg theorems over lattices and other discrete sets
• Quantitative combinatorial geometry for continuous parameters