How to prove the geometric center of a plane shape is inside of the shape if the shape has a convex boundary without hole? How to prove the geometric center of a plane shape is inside of the shape if the shape has a convex boundary without hole?
It is obviously true, but how to prove it mathematically?
 A: To answer this question we need to clarify all definition:
Centroid: arithmetic mean of all points inside a polygon
Convex polygon: a polygon which all segment lines connecting each two points inside the polygon never goes out of polygon.
How do you calculate the centroid? Do you consider each point or, you divide the polygon into many tiny surfaces and calculate their average?
How do you calculate average of: 1,2,3,4,5,6,7,8 ?
You can add all and divide by 8. 
However, since $8=2^n$ an alternative way is to calculate average of each pair:
{1,3}, {2,4}, {5,7}, {6,8}
which are:
2,3,6,7
again I take average of each pair 
{2,6},{3,7}
which are 
4, 5
and the average of the last pair is 4.5 exactly what we expected.
For the polygon, we can calculate the centroid by calculating the average pair by pair. I divide the polygon into $2^n$ pieces where n is large enough. Each piece is very small like a point. I can calculate middle point of that. Since the middle point of each pair is on the line segment connecting them to each other and this line segment must be inside the polygon (since the polygon is convex), I can say the arithmetic mean of each pair is inside the polygon and similarly average mean of next averaged pairs are also inside the polygon and finally the centroid as the last average is inside the polygon too! 
