Continuous quantitative Helly-type results
Tom\'as Fernandez Vidal, Daniel Galicer, Mariano Merzbacher
TL;DR
This paper improves quantitative Helly-type theorems by leveraging a recent sparsification technique, making the results more sensitive to the number of convex sets involved.
Contribution
It introduces a stronger sparsification method to obtain Helly-type theorems that depend on the number of convex sets, enhancing previous results.
Findings
Helly-type theorems now sensitive to the number of sets
Utilizes Friedland and Youssef's recent sparsification result
Improves upon Brazitikos' and Srivastava's earlier work
Abstract
Brazitikos' results on quantititative Helly-type theorems (for the volume and for the diameter) rely on the work of Srivastava on sparsification of John's decompositions. We change this technique by a stronger recent result due to Friedland and Youssef. This, together with an appropriate selection in the accuracy of the approximation, allow us to obtain Helly-type versions which are sensitive to the number of convex sets involved.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsNumerical methods in inverse problems · Optimization and Variational Analysis · Point processes and geometric inequalities