# Jerusalem Mathematics Colloquium

Thursday, 15th January 2009, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Noa Nitzan

(Hebrew University of Jerusalem)

"A planar 3-convex set is indeed a union of six convex sets"

** Abstract: **

Suppose *S* is a planar set. Two points *a,b* in *S*
'see each other' via *S* if *[a,b]* is included in *S*.
F. Valentine proved in 1957 that if *S* is closed, and if for every
three points of *S*, at least two see each other via *S*, then
*S* is a union of three convex sets. The pentagonal star shows that
the number three is best possible.
We discard the condition that *S* is closed and show that *S*
is a union of (at most) six convex sets. The number six is best possible.

Light refreshments will be served in the faculty lounge at 3:30.

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