MathDB

I'd really appreciate help on this.
(a) Given a set

X

of points in the plane, let

f_{X}(n)

be the largest possible area of a polygon with at most

n

vertices, all of which are points of

X

. Prove that if

m, n

are integers with

m \geq n > 2

then

f_{X}(m) + f_{X}(n) \geq f_{X}(m + 1) + f_{X}(n - 1)

.
(b) Let

P_0

be a

1 \times 2

rectangle (including its interior) and inductively define the polygon

P_i

to be the result of folding

P_{i-1}

over some line that cuts

P_{i-1}

into two connected parts. The diameter of a polygon

P_i

is the maximum distance between two points of

P_i

. Determine the smallest possible diameter of

P_{2013}

Problem Statement