I already proved that for any convex polygon and any slope there is a line with this slope that divides the polygon's area in halves. I guess this should help me with the main problem. While proving the lemma I showed that a function of area above the line, when the line moves vertically is continuous and used the intermediate value theorem.
I guess the main problem will be solved in a similar manner. My idea is to take any line dividing the area in halves and start going through all the lines of this property, but changing their slopes. When I'll have rotated 180 degrees I must have gone through a stage when the perimeter was also halved.
If the above reasoning is correct I only need to prove that the function that gives perimeter above the line is continuous when the slope changes.
How do I do this?