Center of mass of convex 3-D shape

Question

Could you please help me, or give the hint:

Let $T\subset \mathbb{R}^{3}$ be a convex 3-D shape, and let $k$ be the center of mass of this shape.

How to prove that $k\in T$?

Thanks a lot!

Answer 1

Assume ${\rm vol}(T)>0$. If the centroid $c$ of $T$ is $\notin T$ then there is a separating hyperplane, which we may assume to be the plane $x_3=0$. We then have $c=(c_1,c_2,c_3)$ with $c_3\leq0$, but all $x\in T$ satisfy $x_3\geq0$. There is even a small ball $B\subset T$ lying completely above the plane $x_3=0$.

Now the formula for the centroid $c$ says that $$c_3={{\displaystyle\int_T x_3\>{\rm dvol}(x)}\over{\rm vol}(T)}\quad .$$ Here the right hand side is $>0$ – a contradiction.