We give a simpler proof of the result of Boros and Füredi
that for any finite set of points in the plane in general
position there is a point lying in 2/9
of all the triangles determined by these points.

Every set *P* of *n* points in **R**^{d} in
general position determines
*d*-simplices. Let *p* be another point in
**R**^{d}. Let *C*(*P*,*p*) be the number of the simplices containing *p*.
Boros and Füredi [2] constructed a set *P* of *n* points in **R**^{2} for which
*C*(*P*,*p*)≤ for every point *p*.
They also proved that there is always a point *p* for which
*C*(*P*,*p*)≥ for every point *p*. Here we present
a new simpler proof of the existence of such a point *p*.

Let *P* be a set of *n* points in the plane.
By the extension of a theorem of Buck and Buck [3] due
to Ceder [4] there are three concurrent lines
that divide the plane into *6* parts each containing at
least *n*/*6*-*1* points in its interior. Denote by *p* the point of
intersection of the three lines. Every choice of six points,
one from each of the six parts, determines a hexagon
containing the point *p*.

Figure 1: a)p∈ABE or p∈BCE | b)p∈ACE and p∈BDF |

Among the triangles determined by the vertices of the hexagon, at
least *8* triangles contain the point *p*. Indeed, from each of
the six pairs of triangles situated as in
Figure 1a we get one triangle
containing *p*. In addition, *p* is contained in both triangles
of the Figure 1b. Therefore, by double
counting, the number of triangles containing *p* is at least

For the sake of completeness we include a sketch of a proof of the modification of the theorem of Buck and Buck that we used above.

**Proposition 1.*** Let μ be a finite measure absolutely continuous with respect to
the Lebesgue measure on R^{2}. Then there are
three concurrent lines that partition the plane into
six parts of equal measure.*

The partition theorem for the finite set of point *P* follows
by letting μ be the restriction of the Lebesgue measure
to the union of tiny disks of equal size centered at the
points of *P*. Since *P*
is in general position, none of the three lines passes through
more than two of the disks.

*Proof sketch.* The given measure can be made into one which gives every open
set a strictly positive measure, and which differs little from
the given one. Proving the result for the latter, and using a
compactness argument, one is through. Hence we can assume the
property mentioned, and we normalize the total measure of the
plane to *1*.

Figure 2: Six rays |

Let now *u* be a unit vector. There is a unique directed line
*L*(*u*)
pointing in the direction *u* and cutting the plane in two parts of
measure *1*/*2*. For any point *P* on *L*(*u*)
there are six unique rays from *P*, denoted *A*(*u*,*P*),…,*F*(*u*,*P*) in clockwise order, splitting the
plane in sectors of measure
*1*/*6*, with *A*(*u*,*P*) in the direction *u*. Note that *L*(*u*) is the union of
*A*(*u*,*P*) and *D*(*u*,*P*). When *P* moves along *L*(*u*) in the direction *u*,
the ray *B*(*u*,*P*)
will turn counterclockwise in a continuous way, becoming orthogonal
to *L*(*u*) at some point. As the clockwise turning *E*(*u*,*P*) behaves in
the same way, there will be a unique *P**(*u*) such that *B*(*u*,*P**(*u*)) and
*E*(*u*,*P**(*u*)) form a line.

The line *L*, the point *P** and the six rays from *P** clearly depend
continuously on *u*. In particular the angle
φ(*u*) one must turn *C*(*u*,*P**(*u*))
counterclockwise to complete *F*(*u*,*P**) to a line varies continuously.
But for any *u*, we have *C*(-*u*,*P**(-*u*))=*F*(*u*,*P**(*u*)), and hence
φ(-*u*)=-φ(*u*). This shows that for some *v* the angle
φ(*v*) vanishes and the rays
*C*(*v*,*P**(*v*)) and *F*(*v*,*P**(*v*)) form a line. This finishes the proof. □

For no dimension higher than *2* the optimal bounds for *C*(*P*,*p*) are
known. Bárány [1] showed that
there is always a point *p* for
which *C*(*P*,p)≥.

*Acknowledgement.* I thank the referee for comments that resulted
in much improved proof of proposition 1.

- 1
- I. Bárány, A generalization of Carathéodory's theorem,
*Discrete Math.***40**(1982), 141–152. - 2
- E. Boros and Z. Füredi, The number of triangles covering the center of an
*n*-set,*Geom. Dedicate***17**(1984), 69–77. - 3
- R. C. Buck and E. F. Buck, Equipartition of convex sets,
*Math. Mag***22**(1949), 195–198. - 4
- J. G. Ceder, Generalized sixpartite problems,
*Bol. Soc. Mat. Mexicana (2)***9**(1964), 28–32.