Open Problem 67
Fair Partitioning of Convex Polygons
From the Open Problems Project
http://maven.smith.edu/~orourke/TOPP/
Nicolas Castet
Computational Geometry CSCE 620 Presentation
Problem definition
 
 
A fair partitioning of a polygon: is a partition of it into a finite number of
pieces so that every piece has both the same area and the same
perimeter. If all the resulting pieces are convex, call it a fair convex
partitioning.
Problem: Given any positive integer n, can any convex polygon be
convex fair partitioned into n pieces?
Posed by R. Nandakumar and N. Ramana Rao, June 2007.
 
This problem is related to the Cake Cutting problem
which does not have any constraint on the perimeter
H. Steinhaus, The problem of fair division, Econometrica 16 (1948)
101-104
A regular hexagon convex
fair partitioned into 7 pieces.
Fair Partitioning of Convex Polygons
2
Results (n=2)
 
Fair convex partitioning of a convex polygon
into n=2 pieces
–  Proof:
 
 
 
 
 
line P-Q divides the polygon into two equal area convex pieces
Suppose piece 1 has a larger perimeter than 2.
Move P clockwise keeping the two areas equal (Q is also moving, but
maybe not at the same speed) until 2 pieces have equal perimeters.
If we move P until reaching the initial position of Q, 2 has now a larger
perimiter than 1.
By continuity, there is a solution
P
1
P’
2
Q’
R. Nandakumar and N. Ramana Rao
'Fair' partitions of polygons--an introduction, 2008
Fair Partitioning of Convex Polygons
Q
3
Results (n=4)
 
Fair convex partitioning of a convex polygon
into n=4 pieces
–  We can’t apply recursively previous proof to this case
–  Proof:
 
 
 
 
Divide polygon into two equal area pieces by any area bisector
Consider a fair bisector of piece A, that divides A into pieces{A1, A2}
and likewise B into pieces {B1,B2}
Rotate the area bisector keeping the areas of
A and B equal and the 2 fair bisectors move
too to keep their properties
Like the previous proof, by continuity
there is a solution.
R. Nandakumar and N. Ramana Rao
'Fair' partitions of polygons--an introduction, 2008
Fair Partitioning of Convex Polygons
4
Result (n=2k, and n=3)
 
Generalization n=4 to all power of 2, n=2k pieces
R. Nandakumar and N. Ramana Rao
'Fair' partitions of polygons--an introduction, 2008
 
n=3 pieces
–  Sophisticated proof using methods from equivariant
topology
Barany, P. Blagojevic and A. Szucs. ‘Equipartitioning by a Convex 3-fan’.
Advances in Mathematics, Volume 223, Issue 2, January 2010, pages 579-593.
Fair Partitioning of Convex Polygons
5
Result (n=pk, p a prime number)
 
n=pk pieces, with p a prime number
–  Sophisticated proof
–  Proof works for higher dimensions
B. Aronov, A. Hubard. Convex equipartitions of volume and surface area. 2010
R.N. Karasev, Equipartition of several measures, 2010
P. Soberon, Balanced convex partitions of measures in Rd, 2010
Fair Partitioning of Convex Polygons
6
Result summary
 
 
Problem: Given any positive integer n, can any
convex polygon be convex fair partitioned into n
pieces?
Proof:
–  n=2, n=4, and n=2k
–  n=3
2008, R. Nandakumar and N. Ramana Rao
–  n=pk, p a prime number
2010, B. Aronov, A. Hubard
2010, Barany, P. Blagojevic and A. Szucs
2010, R.N. Karasev
2010, P. Soberon
 
Weaker result:
–  Any polygon allows fair partitioning for any n (where the pieces need not be
convex) 2006, R. Nandakumar
Fair Partitioning of Convex Polygons
7
Future directions
 
 
 
 
“the originators tend to believe every convex polygon allows a
fair convex partition into n pieces for any n”
There is no proof for the general case with n, a positive integer.
There is no work on algorithms for the fair convex partitioning of
convex polygons and the fair partitioning of any polygon.
If there is “not always” a fair convex partition, how does one
decide the possibility of such a partitioning for a given polygon
and a given n?
And if a fair convex partition exists for a specific polygon, how
does one find a fair partitioning that minimizes the sum of the
perimeters of the pieces?
Fair Partitioning of Convex Polygons
8
Applications
 
 
“Imagine that you are cooking chicken at a party. You will cut
the raw chicken fillet with a sharp knife, marinate each of the
pieces in a spicy sauce and then fry the pieces. The surface of
each piece will be crispy and spicy. Can you cut the chicken so
that all your guests get the same amount of crispy crust and the
same amount of chicken?” J 2010, B. Aronov, A. Hubard
Our problem is similar to that considering two-dimensional
convex chickens.
Other application: Special discretization of spatial domains
Fair Partitioning of Convex Polygons
9
References
http://maven.smith.edu/~orourke/TOPP/P67.html
H. Steinhaus, The problem of fair division, Econometrica 16 (1948)
101-104
R. Nandakumar and N. Ramana Rao
'Fair' partitions of polygons--an introduction, 2008
Barany, P. Blagojevic and A. Szucs. ‘Equipartitioning by a Convex 3-fan’.
Advances in Mathematics, Volume 223, Issue 2, January 2010, pages 579-593.
B. Aronov, A. Hubard. Convex equipartitions of volume and surface area. 2010
R.N. Karasev, Equipartition of several measures, 2010
P. Soberon, Balanced convex partitions of measures in Rd, 2010
Fair Partitioning of Convex Polygons
10