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On F-convexity and related problems. Yuan, Liping
Description
If every $k$-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property $T(k)$. We say that a family $\mathcal{F}$ has property $T-m$, if there exists a subfamily $\mathcal{G} \subset \mathcal{F}$ with $|\mathcal{F} - \mathcal{G}| \le m$ admitting a line transversal. In 2007, Heppes posed the problem whether there exists a convex body $K$ in the plane such that if $\mathcal{F}$ is a finite $T(3)$-family of disjoint translates of $K$, then $m=3$ is the smallest value for which $\mathcal{F}$ has property $T-m$. In this talk, we study this open problem {in terms of} finite $T(3)$-families of pairwise disjoint translates of a regular $2n$-gon $(n \ge 5)$. We find out that, for $5 \le n \le 34$, the family has property $T - 3$; for $n \ge 35$, the family has property $T - 2$. This is a joint work with Qingdan Du and Tudor Zamfirescu.
Item Metadata
Title |
On F-convexity and related problems.
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-10-08T11:00
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Description |
If every $k$-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property $T(k)$. We say that a family $\mathcal{F}$ has property $T-m$, if there
exists a subfamily $\mathcal{G} \subset \mathcal{F}$ with $|\mathcal{F}
- \mathcal{G}| \le m$ admitting a line
transversal. In 2007, Heppes posed the problem whether there exists a convex body $K$
in the plane such that if $\mathcal{F}$ is a finite $T(3)$-family of disjoint translates of $K$,
then $m=3$ is the smallest value for which $\mathcal{F}$ has property $T-m$. In this talk,
we study this open problem {in terms of} finite $T(3)$-families of pairwise disjoint translates of
a regular $2n$-gon $(n \ge 5)$. We find out that, for $5 \le n \le 34$, the family has property $T - 3$;
for $n \ge 35$, the family has property $T - 2$.
This is a joint work with Qingdan Du and Tudor Zamfirescu.
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Extent |
41.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Hebei Normal Universtiy
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Series | |
Date Available |
2020-04-06
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0389742
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International