- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- BIRS Workshop Lecture Videos /
- Colorful coverings of polytopes -- the hidden topological...
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
Colorful coverings of polytopes -- the hidden topological truth behind different colorful phenomena Zerbib, Shira
Description
The topological KKMS Theorem is a powerful extension of Brouwer's Fixed-Point Theorem, which was proved by Shapley in 1973 in the context of game theory. We prove a colorful and polytopal generalization of the KKMS Theorem, and show that our theorem implies some seemingly unrelated results in discrete geometry and combinatorics involving colorful settings. For example, we apply our theorem to provide a new proof of the Colorful Caratheodory Theorem due to Barany, and also to obtain an upper bound on the piercing numbers in colorful d-interval families, extending results of Tardos, Kaiser and Alon for the non-colored case. We further apply our theorem to questions regarding envy-free fair division of goods among a set of players. Joint with Florian Frick.
Item Metadata
Title |
Colorful coverings of polytopes -- the hidden topological truth behind different colorful phenomena
|
Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
|
Date Issued |
2018-02-07T10:40
|
Description |
The topological KKMS Theorem is a powerful extension of Brouwer's Fixed-Point Theorem, which was proved by Shapley in 1973 in the context of game theory. We prove a colorful and polytopal generalization of the KKMS Theorem, and show that our theorem implies some seemingly unrelated results in discrete geometry and combinatorics involving colorful settings.
For example, we apply our theorem to provide a new proof of the Colorful Caratheodory Theorem due to Barany, and also to obtain an upper bound on the piercing numbers in colorful d-interval families, extending results of Tardos, Kaiser and Alon for the non-colored case. We further apply our theorem to questions regarding envy-free fair division of goods among a set of players.
Joint with Florian Frick.
|
Extent |
35 minutes
|
Subject | |
Type | |
File Format |
video/mp4
|
Language |
eng
|
Notes |
Author affiliation: University of Michigan
|
Series | |
Date Available |
2018-08-07
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0369721
|
URI | |
Affiliation | |
Peer Review Status |
Unreviewed
|
Scholarly Level |
Researcher
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International