MathDB

For a set

S

of at least

3

points in the plane, let

d_{\text{min}}

denote the minimal distance between two different points in

S

and

d_{\text{max}}

the maximal distance between two different points in

S

.
For a real

c>0

, a set

S

will be called

c

-balanced if

\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|

Prove that there exists a real

c>0

so that for every

c

-balanced set of points

S

, there exists a triangle with vertices in

S

that contains at least

\sqrt{|S|}

elements of

S

in its interior or on its boundary.

Problem Statement