Problem 6

Assign to each side

b

of a convex polygon

P

the maximum area of a triangle that has

b

as a side and is contained in

P

. Show that the sum of the areas assigned to the sides of

P

is at least twice the area of

P

Step 2 of 5: Pigeonhole gives a butterfly of area at least

S/m

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\sum_{i=0}^{m-1}[B_i]\ge S\ \Longrightarrow\ \exists\,i:\ [B_i]\ge\frac{S}{m}

Detailed analysis

Since the $m$ butterflies $B_0,\dots,B_{m-1}$ cover the polygon $Q$ of area $S$ , the sum of their areas is at least $S$ . By averaging, at least one butterfly $B_i$ has area $[B_i]\ge S/m$ .